Ncert Solutions Conic Sections Class 11
Exercise 11.1
Question 1:
Find the equation of the circle with centre (0, 2) and radius 2.
Answer:
The equation of a circle with centre (h, k) and radius r is given as
(x – h)2 + (y – k)2 = r2
It is given that centre (h, k) = (0, 2) and radius (r) = 2.
Therefore, the equation of the circle is
(x – 0)2 + (y – 2)2 = 22
=> x2 + y2 + 4 – 4y = 4
=> x2 + y2 – 4y = 0
Question 2:
Find the equation of the circle with centre (–2, 3) and radius 4.
Answer:
The equation of a circle with centre (h, k) and radius r is given as
(x – h)2 + (y – k)2 = r2
It is given that centre (h, k) = (–2, 3) and radius (r) = 4.
Therefore, the equation of the circle is
(x + 2)2 + (y – 3)2 = (4)2
=> x2 + 4x + 4 + y2 – 6y + 9 = 16
=> x2 + y2 + 4x – 6y – 3 = 0
Question 3:
Find the equation of the circle with centre (1/2, 1/4) and radius 1/12.
Answer:
The equation of a circle with centre (h, k) and radius r is given as
(x – h)2 + (y – k)2 = r2
It is given that centre (h, k) = (1/2, 1/4) and radius (r) = 1/12
Therefore, the equation of the circle is
(x – 1/2)2 + (y – 1/4)2 = (1/12)2
=> x2 – x + 1/4 + y2 – y/2 + 1/16 = 1/144
=> x2 – x + 1/4 + y2 – y/2 + 1/16 - 1/144 = 0
=> (144x2 – 144x + 36 + 144y2 – 72y + 9 – 1)/144 = 0
=> 144x2 – 144x + 36 + 144y2 – 72y + 9 – 1 = 0
=> 144x2 – 144x + 144y2 – 72y + 44 = 0
=> 4(36x2 – 36x + 36y2 – 18y + 11) = 0
=> 36x2 – 36x + 36y2 – 18y + 11 = 0
=> 36x2 + 36y2 - 36x – 18y + 11 = 0
Question 4:
Find the equation of the circle with centre (1, 1) and radius √2.
Answer:
The equation of a circle with centre (h, k) and radius r is given as
(x – h)2 + (y – k)2 = r2
It is given that centre (h, k) = (1, 1) and radius (r) = √2.
Therefore, the equation of the circle is
(x - 1)2 + (y - 1)2 = (√2)2
=> x2 – 2x + 1 + y2 – 2y + 1 = 2
=> x2 + y2 –2x - 2y = 0
Question 5:
Find the equation of the circle with centre (–a, –b) and radius √(a2 − b2).
Answer:
The equation of a circle with centre (h, k) and radius r is given as
(x – h)2 + (y – k)2 = r2
It is given that centre (h, k) = (–a, –b) and radius (r) = √(a2 − b2)
Therefore, the equation of the circle is
(x + a)2 + (y + b)2 = {√(a2 − b2)}2
=> x2 + 2ax + a2 + y2 + 2by + b2 = a2 − b2
=> x2 + y2 + 2ax + 2by + 2b2 = 0
Question 6:
Find the centre and radius of the circle (x + 5)2 + (y – 3)2 = 36
Answer:
The equation of the given circle is (x + 5)2 + (y – 3)2 = 36.
(x + 5)2 + (y – 3)2 = 36
=> {x – (–5)}2 + (y – 3)2 = 62
Which is of the form (x – h)2 + (y – k)2 = r2
Where h = – 5, k = 3, and r = 6.
Thus, the centre of the given circle is (–5, 3) and its radius is 6.
Question 7:
Find the centre and radius of the circle x2 + y2 – 4x – 8y – 45 = 0
Answer:
The equation of the given circle is x2 + y2 – 4x – 8y – 45 = 0
=> x2 + y2 – 4x – 8y – 45 = 0
=> (x2 – 4x) + (y2 – 8y) = 45
=> {x2 – 2 * x * 2 + 22} + {y2 – 2 * y * 4 + 42} – 4 –16 = 45
=> (x – 2)2 + (y –4)2 = 65
=> (x – 2)2 + (y –4)2 = (√65)2
Which is of the form (x – h)2 + (y – k)2 = r2
Where h = 2, k = 4 and r = √65
Thus, the centre of the given circle is (2, 4), while its radius is √65
Question 8:
Find the centre and radius of the circle x2 + y2 – 8x + 10y – 12 = 0
Answer:
The equation of the given circle is x2 + y2 – 8x + 10y – 12 = 0
=> x2 + y2 – 8x + 10y – 12 = 0
=> (x2 – 8x) + (y2 + 10y) = 12
=> {x2 – 2 * x * 4 + 42} + {y2 + 2 * y * 5 + 52} – 16 –25 = 12
=> (x – 4)2 + (y + 5)2 = 53
=> (x – 4)2 + (y + 5)2 = (√53)2
=> (x – 4)2 + {y – (-5)}2 = (√53)2
Which is of the form (x – h)2 + (y – k)2 = r2
Where h = 4, k = -5 and r = √53
Thus, the centre of the given circle is (4, -5), while its radius is √53.
Question 9:
Find the centre and radius of the circle 2x2 + 2y2 – x = 0
Answer:
The equation of the given circle is 2x2 + 2y2 – x = 0
=> 2x2 + 2y2 – x = 0
=> 2x2 – x + 2y2 = 0
=> 2(x2 – x/2 + y2) = 0
=> x2 – x/2 + y2 = 0
=> x2 – 2 * x * (1/4) + (1/4)2 + y2 - (1/4)2 = 0
=> (x – 1/4)2 + (y – 0)2 = (1/4)2
which is of the form (x – h)2 + (y – k)2 = r2
where h = 1/4, k = 0 and r = 1/4
Thus, the centre of the given circle is (1/4, 0), while its radius is 1/4.
Question 10:
Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.
Answer:
Let the equation of the required circle be (x – h)2 + (y – k)2 = r2
Since the circle passes through points (4, 1) and (6, 5),
(4 – h)2 + (1 – k)2 = r2 ...................... 1
(6 – h)2 + (5 – k)2 = r2 ...................... 2
Since the centre (h, k) of the circle lies on line 4x + y = 16,
4h + k = 16 .....................3
From equations (1) and (2), we obtain
(4 – h)2 + (1 – k)2 = (6 – h)2 + (5 – k)2
=> 16 – 8h + h2 + 1 – 2k + k2 = 36 – 12h + h2 + 25 – 10k + k2
=> 16 – 8h + 1 – 2k = 36 – 12h + 25 – 10k
=> 4h + 8k = 44
=> h + 2k = 11 ..................4
On solving equations 3 and 4, we obtain h = 3 and k = 4.
On substituting the values of h and k in equation 1, we obtain
(4 – 3)2 + (1 – 4)2 = r2
=> (1)2 + (– 3)2 = r2
=> 1 + 9 = r2
=> r2 = 10
=> r = √10
Thus, the equation of the required circle is
(x – 3)2 + (y – 4)2 = (√10)2
=> x2 – 6x + 9 + y2 – 8y + 16 = 10
=> x2 + y2 – 6x – 8y + 15 = 0
Question 11:
Find the equation of the circle passing through the points (2, 3) and (–1, 1) and whose centre is on the line x – 3y – 11 = 0.
Answer:
Let the equation of the required circle be (x – h)2 + (y – k)2 = r2
Since the circle passes through points (2, 3) and (–1, 1),
(2 – h)2 + (3 – k)2 = r2 ...................1
(–1 – h)2 + (1 – k)2 = r2 .................2
Since the centre (h, k) of the circle lies on line x – 3y – 11 = 0,
So, h – 3k = 11 .....................3
From equations 1 and 2, we obtain
(2 – h)2 + (3 – k)2 = (–1 – h)2 + (1 – k)2
=> 4 – 4h + h2 + 9 – 6k + k2 = 1 + 2h + h2 + 1 – 2k + k2
=> 4 – 4h + 9 – 6k = 1 + 2h + 1 – 2k
=> 6h + 4k = 11 ................4
On solving equations (3) and (4), we obtain h = 7/2 and k = −5/2
On substituting the values of h and k in equation 1, we obtain
(2 – 7/2)2 + (3 + 5/2)2 = r2
=> (-3/2)2 + (11/2)2 = r2
=> 9/4 + 121/4 = r2
=> r2 = 130/4
Thus, the equation of the required circle is
(x – 7/2)2 + (y + 5/2)2 = 130/4
=> {(2x – 7)/2}2 + {(2y + 5)/2}2 = 130/4
=> 4x2 – 28x + 49 + 4y2 + 20y + 25 = 130
=> 4x2 – 28x + 49 + 4y2 + 20y + 25 – 130 = 0
=> 4x2 – 28x + 4y2 + 20y - 56 = 0
=> 4(x2 + y2 – 7x + 5y – 14) = 0
=> x2 + y2 – 7x + 5y – 14 = 0
Question 12:
Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2, 3).
Answer:
Let the equation of the required circle be (x – h)2 + (y – k)2 = r2
Since the radius of the circle is 5 and its centre lies on the x-axis, k = 0 and r = 5.
Now, the equation of the circle becomes (x – h)2 + y2 = 25
It is given that the circle passes through point (2, 3).
So, (2 – h)2 + 32 = 25
=> (2 – h)2 + 9 = 25
=> (2 – h)2 = 25 - 9
=> (2 – h)2 = 16
=> 2 – h = ±√16
=> 2 – h = ±4
If 2 – h = 4, then h = -2
If 2 – h = -4, then h = 6
When h = –2, the equation of the circle becomes
(x + 2)2 + y2 = 25
=> x2 + 4x + 4 + y2 = 25
=> x2 + y2 + 4x – 21 = 0
When h = 6, the equation of the circle becomes
(x – 6)2 + y2 = 25
=> x2 – 12x + 36 + y2 = 25
=> x2 + y2 – 12x + 11 = 0
Question 13:
Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.
Answer:
Let the equation of the required circle be (x – h)2 + (y – k)2 = r2
Since the centre of the circle passes through (0, 0),
(0 – h)2 + (0 – k)2 = r2
=> h2 + k2 = r2
The equation of the circle now becomes (x – h)2 + (y – k)2 = h2 + k2
It is given that the circle makes intercepts a and b on the coordinate axes. This means that the
circle passes through points (a, 0) and (0, b). Therefore,
(a – h)2 + (0 – k)2 = h2 + k2 ........................ (1)
(0 – h)2 + (b – k)2 = h2 + k2 ........................ (2)
From equation (1), we obtain
a2 – 2ah + h2 + k2 = h2 + k2
=> a2 – 2ah = 0
=> a(a – 2h) = 0
=> a = 0 or (a – 2h) = 0
However, a ≠ 0; hence, (a – 2h) = 0
=> h =a/2
From equation (2), we obtain
h2 + b2 – 2bk + k2 = h2 + k2
=> b2 – 2bk = 0
=> b(b – 2k) = 0
=> b = 0 or(b – 2k) = 0
However, b ≠ 0; hence, (b – 2k) = 0
=> k = b/2
Thus, the equation of the required circle is
(x – a/2)2 + (y – b/2)2 = (a/2)2 + (b/2)2
=> {(2x – a)/2}2 + {(2y – b)/2}2 = a2/4 + b2/4
=> (4x2 – 4ax + a2)/4 + (4y2 – 4by + b2)/4 = (a2 + b2)/4
=> 4x2 – 4ax + a2 + 4y2 – 4by + b2 = a2 + b2
=> 4x2 + 4y2 – 4ax – 4by = 0
=> 4(x2 + y2 – ax – by) = 0
=> x2 + y2 – ax – by = 0
Question 14:
Find the equation of a circle with centre (2, 2) and passes through the point (4, 5).
Answer:
The centre of the circle is given as (h, k) = (2, 2)
Since the circle passes through point (4, 5), the radius (r) of the circle is the distance between
the points (2, 2) and (4, 5).
So, r = √{(2 - 4)2 + (2 - 5)2} = √{(-2)2 + (-3)2} = √(4 + 9) = √13
Thus, the equation of the circle is
(x – h)2 + (y – k)2 = r2
=> (x – 2)2 + (y – 2)2 = (√13)2
=> x2 – 4x + 4 + y2 – 4y + 4 = 13
=> x2 + y2 – 4x – 4y - 5 = 0
Question 15:
Does the point (–2.5, 3.5) lie inside, outside or on the circle x2 + y2 = 25?
Answer:
The equation of the given circle is x2 + y2 = 25
=> x2 + y2 = 25
=> (x – 0)2 + (y – 0)2 = 52
which is of the form (x – h)2 + (y – k)2 = r2
where h = 0, k = 0, and r = 5.
So, Centre = (0, 0) and radius = 5
Now, distance between point (–2.5, 3.5) and centre (0, 0)
= √{(-2.5 - 0)2 + (3.5 - 0)2}
= √{(-2.5)2 + (3.5)2}
= √(6.25 + 12.25)
= √18.5
= 4.3 < 5
Since the distance between point (–2.5, 3.5) and centre (0, 0) of the circle is less than the
radius of the circle,
So, the point (-2.5, 3.5) lies inside the circle.
Exercise 11.2
Question 1:
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y2 = 12x.
Answer:
The given equation is y2 = 12x.
Here, the coefficient of x is positive. Hence, the parabola opens towards the right.
On comparing this equation with y2 = 4ax, we obtain
4a = 12
=> a = 3
So, the coordinates of the focus = (a, 0) = (3, 0)
Since the given equation involves y2, the axis of the parabola is the x-axis.
Equation of direcctrix, x = –a i.e., x = – 3 i.e., x + 3 = 0
Length of latus rectum = 4a = 4 * 3 = 12
Question 2:
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x2 = 6y.
Answer:
The given equation is x2 = 6y.
Here, the coefficient of y is positive. Hence, the parabola opens upwards.
On comparing this equation with x2 = 4ay, we obtain
4a = 6
=> a = 6/4
=> a = 3/2
So, the coordinates of the focus = (0, a) = (0, 3/2)
Since the given equation involves x2, the axis of the parabola is the y-axis.
Equation of directrix, y = -a i.e. y = -3/2
Length of latus rectum = 4a = 4 * 3/2 = 2 * 3 = 6
Question 3:
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y2 = – 8x.
Answer 3:
The given equation is y2 = –8x.
Here, the coefficient of x is negative. Hence, the parabola opens towards the left.
On comparing this equation with y2 = –4ax, we obtain
–4a = –8
=> 4a = 8
=> a = 2
So, the coordinates of the focus = (–a, 0) = (–2, 0)
Since the given equation involves y2, the axis of the parabola is the x-axis.
Equation of directrix, x = a i.e., x = 2
Length of latus rectum = 4a = 8
Question 4:
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x2 = – 16y.
Answer:
The given equation is x2 = –16y.
Here, the coefficient of y is negative. Hence, the parabola opens downwards.
On comparing this equation with x2 = – 4ay, we obtain
–4a = –16
=> 4a = 16
=> a = 4
So, the coordinates of the focus = (0, –a) = (0, –4)
Since the given equation involves x2, the axis of the parabola is the y-axis.
Equation of directrix, y = a i.e., y = 4
Length of latus rectum = 4a = 16
Question 5:
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y2 = 10x.
Answer:
The given equation is y2 = 10x.
Here, the coefficient of x is positive. Hence, the parabola opens towards the right.
On comparing this equation with y2 = 4ax, we obtain
4a = 10
=> a = 10/4
=> a = 5/2
Coordinates of the focus = (a, 0) = (5/2, 0)
Since the given equation involves y2, the axis of the parabola is the x-axis.
Equation of directrix, x = -a i.e. x = -5/2
Length of latus rectum = 4a = 10
Question 6:
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x2 = –9y.
Answer:
The given equation is x2 = –9y.
Here, the coefficient of y is negative. Hence, the parabola opens downwards.
On comparing this equation with x2 = –4ay, we obtain
-4a = -9
=> 4a = 9
=> a = 9/4
So, the coordinates of the focus = (0, -a) = (0, -9/4)
Since the given equation involves x2, the axis of the parabola is the y-axis.
Equation of directrix, y = a i.e. y = 9/4
Length of latus rectum = 4a = 9
Question 7:
Find the equation of the parabola that satisfies the following conditions: Focus (6, 0); directrix x = –6.
Answer:
Given, Focus is (6, 0) and the directrix, x = –6
Since the focus lies on the x-axis, the x-axis is the axis of the parabola.
Therefore, the equation of the parabola is either of the form y2 = 4ax or y2 = – 4ax.
It is also seen that the directrix, x = –6 is to the left of the y-axis, while the focus (6, 0) is to the
right of the y-axis.
Hence, the parabola is of the form y2 = 4ax.
Here, a = 6
Thus, the equation of the parabola is y2 = 24x.
Question 8:
Find the equation of the parabola that satisfies the following conditions: Focus (0, –3); directrix y = 3.
Answer:
Given, Focus = (0, –3) and the directrix y = 3
Since the focus lies on the y-axis, the y-axis is the axis of the parabola.
Therefore, the equation of the parabola is either of the form x2 = 4ay or x2 = – 4ay.
It is also seen that the directrix, y = 3 is above the x-axis, while the focus (0, –3) is below the
x-axis.
Hence, the parabola is of the form x2 = –4ay.
Here, a = 3
Thus, the equation of the parabola is x2 = –12y.
Question 9:
Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0); focus (3, 0).
Answer:
Given, Vertex (0, 0) and focus (3, 0)
Since the vertex of the parabola is (0, 0) and the focus lies on the positive x-axis, x-axis is the
axis of the parabola, while the equation of the parabola is of the form y2 = 4ax.
Since the focus is (3, 0),
So, a = 3
Thus, the equation of the parabola is y2 = 4 * 3 * x, i.e., y2 = 12x
Question 10:
Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0), focus (–2, 0).
Answer:
Given, Vertex = (0, 0) and focus = (–2, 0)
Since the vertex of the parabola is (0, 0) and the focus lies on the negative x-axis, x-axis is the
axis of the parabola, while the equation of the parabola is of the form y2 = – 4ax.
Since the focus is (–2, 0),
So, a = 2
Thus, the equation of the parabola is y2 = –4 * 2 * x, i.e., y2 = –8x
Question 11:
Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) passing through (2, 3) and axis is along x-axis.
Answer:
Since the vertex is (0, 0) and the axis of the parabola is the x-axis, the equation of the parabola
is either of the form y2 = 4ax or y2 = –4ax.
The parabola passes through point (2, 3), which lies in the first quadrant.
Therefore, the equation of the parabola is of the form y2 = 4ax,
while point (2, 3) must satisfy the equation y2 = 4ax.
So, 32 = 4a * 2
=> 9 = 8a
=> a = 9/8
Thus, the equation of the parabola is
y2 = 4 * (9/8) * x
=> y2 = 9x/2
=> 2y2 = 9x
Question 12:
Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis.
Answer:
Since the vertex is (0, 0) and the parabola is symmetric about the y-axis, the equation of the
parabola is either of the form x2 = 4ay or x2 = –4ay.
The parabola passes through point (5, 2), which lies in the first quadrant.
Therefore, the equation of the parabola is of the form x2 = 4ay,
while point (5, 2) must satisfy the equation x2 = 4ay.
So, 52 = 4a * 2
=> 25 = 8a
=> a = 25/8
Thus, the equation of the parabola is
x2 = 4 * (25/8) * y
=> x2 = 25y/2
=> 2x2 = 25y
Exercise 11.3
Question 1:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,
the eccentricity and the length of the latus rectum of the ellipse x2/36 + y2/16 = 1.
Answer:
The given equation is x2/36 + y2/16 = 1 or or x2/62 + y2/42 = 1.
Here, the denominator of x2 is greater than the denominator of y2.
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with x2/a2 + y2/b2 = 1, we get
a = 6 and b = 4.
Now, c = √(a2 - b2) = √(36 - 16) = √20 = 2√5
Therefore,
The coordinates of the foci are (2√5, 0) and (-2√5, 0).
The coordinates of the vertices are (6, 0) and (–6, 0).
Length of major axis = 2a = 12
Length of minor axis = 2b = 8
Eccentricity, e = c/a = 2√5/6 = √5/3
Length of latus rectum = 2b2/a = (2 * 16)/6 = 16/3
Question 2:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,
the eccentricity and the length of the latus rectum of the ellipse x2/4 + y2/25 = 1.
Answer:
The given equation is x2/4 + y2/25 = 1 or or x2/22 + y2/52 = 1.
Here, the denominator of y2 is greater than the denominator of x2.
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with x2/b2 + y2/a2 = 1, we get
b = 2 and a = 5.
Now, c = √(a2 - b2) = √(25 - 4) = √21
Therefore,
The coordinates of the foci are (0, √21) and (0, -√21)
The coordinates of the vertices are (0, 5) and (0, –5)
Length of major axis = 2a = 10
Length of minor axis = 2b = 4
Eccentricity e = c/a = √21/5
Length of latus rectum = 2b2/a = (2 * 4)/5 = 8/5
Question 3:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,
the eccentricity and the length of the latus rectum of the ellipse x2/16 + y2/9 = 1.
Answer:
The given equation is x2/16 + y2/9 = 1 or x2/42 + y2/32 = 1.
Here, the denominator of x2 is greater than the denominator of y2.
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with x2/a2 + y2/b2 = 1, we get
a = 4 and b = 3.
Now, c = √(a2 - b2) = √(16 - 9) = √7
Therefore,
The coordinates of the foci are (±√7, 0)
The coordinates of the vertices are (±4, 0)
Length of major axis = 2a = 8
Length of minor axis = 2b = 6
Eccetricity, e = c/a = √7/4
Length of latus rectum = 2b2/a = (2 * 9)/4 = 9/2
Question 4:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,
the eccentricity and the length of the latus rectum of the ellipse x2/25 + y2/100 = 1.
Answer:
The given equation is x2/25 + y2/100 = 1 or or x2/52 + y2/102 = 1.
Here, the denominator of y2 is greater than the denominator of x2.
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with x2/b2 + y2/a2 = 1, we get
b = 5 and a = 10.
Now, c = √(a2 - b2) = √(100 - 25) = √75 = 5√3
Therefore,
The coordinates of the foci are (0, 5√3) and (0, -5√3)
The coordinates of the vertices are (0, 10) and (0, –10)
Length of major axis = 2a = 20
Length of minor axis = 2b = 10
Eccentricity, e = c/a = 5√3/10 = √3/2
Length of latus rectum = 2b2/a = (2 * 25)/10 = 25/5 = 5
Question 5:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,
the eccentricity and the length of the latus rectum of the ellipse x2/49 + y2/36 = 1.
Answer:
The given equation is x2/49 + y2/36 = 1 or x2/72 + y2/62 = 1.
Here, the denominator of x2 is greater than the denominator of y2.
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with x2/a2 + y2/b2 = 1, we get
a = 7 and b = 6.
Now, c = √(a2 - b2) = √(49 - 36) = √13
Therefore,
The coordinates of the foci are (±√13, 0)
The coordinates of the vertices are (±7, 0)
Length of major axis = 2a = 14
Length of minor axis = 2b = 12
Eccetricity, e = c/a = √13/7
Length of latus rectum = 2b2/a = (2 * 36)/7 = 72/7
Question 6:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity
and the length of the latus rectum of the ellipse x2/100 + y2/400 = 1.
Answer:
The given equation is x2/100 + y2/400 = 1 or or x2/102 + y2/202 = 1.
Here, the denominator of y2 is greater than the denominator of x2.
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with x2/b2 + y2/a2 = 1, we get
b = 10 and a = 20.
Now, c = √(a2 - b2) = √(400 - 100) = √300 = 10√3
Therefore,
The coordinates of the foci are (0, 10√3) and (0, -10√3)
The coordinates of the vertices are (0, 20) and (0, –20)
Length of major axis = 2a = 40
Length of minor axis = 2b = 20
Eccentricity, e = c/a = 10√3/20 = √3/2
Length of latus rectum = 2b2/a = (2 * 100)/20 = 2 * 5 = 10
Question 7:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the
length of the latus rectum of the ellipse 36x2 + 4y2 = 144.
Answer:
The given equation is 36x2 + 4y2 = 144
36x2/144 + 4y2/144 = 1
x2/4 + y2/36 = 1
x2/22 + y2/62 = 1.
Here, the denominator of y2 is greater than the denominator of x2.
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with x2/b2 + y2/a2 = 1,
we get b = 2 and a = 6
Now, c = √(a2 - b2) = √(36 - 4) = √32 = 4√2
Therefore,
The coordinates of the foci are (0, 4√2) and (0, -4√2)
The coordinates of the vertices are (0, 6) and (0, –6)
Length of major axis = 2a = 12
Length of minor axis = 2b = 4
Eccentricity, e = c/a = 4√2/6 = 2√2/3
Length of latus rectum = 2b2/a = (2 * 4)/6 = 4/3
Question 8:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the
length of the latus rectum of the ellipse 16x2 + y2 = 16.
Answer:
The given equation is 16x2 + y2 = 16
16x2/16 + y2/16 = 1
x2/1 + y2/16 = 1
x2/12 + y2/42 = 1.
Here, the denominator of y2 is greater than the denominator of x2.
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with x2/b2 + y2/a2 = 1,
we get b = 1 and a = 4
Now, c = √(a2 - b2) = √(16 - 1) = √15
Therefore,
The coordinates of the foci are (0, √15) and (0, -√15)
The coordinates of the vertices are (0, 4) and (0, –4)
Length of major axis = 2a = 8
Length of minor axis = 2b = 2
Eccentricity, e = c/a = √15/4
Length of latus rectum = 2b2/a = (2 * 1)/4 = 1/2
Question 9:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4x2 + 9y2 = 36.
Answer:
The given equation is 4x2 + 9y2 = 36
4x2/36 + 9y2/36 = 1
x2/9 + y2/4 = 1
x2/32 + y2/22 = 1.
Here, the denominator of x2 is greater than the denominator of y2.
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with x2/a2 + y2/b2 = 1,
we get b = 2 and a = 3
Now, c = √(a2 - b2) = √(9 - 4) = √5
Therefore,
The coordinates of the foci are (±√5, 0) and
The coordinates of the vertices are (±3, 0)
Length of major axis = 2a = 6
Length of minor axis = 2b = 4
Eccentricity, e = c/a = √5/3
Length of latus rectum = 2b2/a = (2 * 4)/3 = 8/3
Question 10:
Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0).
Answer:
Given, Vertices (±5, 0), foci (±4, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the ellipse will be of the form x2/a2 + y2/b2 = 1, where a is the
semi-major axis.
Accordingly, a = 5 and c = 4.
We know that
a2 = b2 + c2
=> 52 = b2 + 42
=> b2 = 25 - 16
=> b2 = 9
=> b = 3
Thus, the equation of the ellipse is x2/52 + y2/32 = 1 or x2/25 + y2/9 = 1
Question 11:
Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5).
Answer:
Given, Vertices (0, ±13), foci (0, ±5).
Here, the vertices are on the y-axis.
Therefore, the equation of the ellipse will be of the form x2/b2 + y2/a2 = 1, where a is the
semi-major axis.
Accordingly, a = 13 and c = 5.
We know that
a2 = b2 + c2
=> 132 = b2 + 52
=> b2 = 169 - 25
=> b2 = 144
=> b = 12
Thus, the equation of the ellipse is x2/122 + y2/132 = 1 or x2/144 + y2/169 = 1
Question 12:
Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0).
Answer:
Given, Vertices (±6, 0), foci (±4, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the ellipse will be of the form x2/a2 + y2/b2 = 1, where a is the
semi-major axis.
Accordingly, a = 6 and c = 4.
We know that
a2 = b2 + c2
=> 62 = b2 + 42
=> b2 = 36 - 16
=> b2 = 20
=> b = √20
Thus, the equation of the ellipse is x2/62 + y2/(√20)2 = 1 or x2/36 + y2/20 = 1
Question 13:
Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3, 0), ends of minor axis (0, ±2).
Answer:
Given, Ends of major axis (±3, 0), ends of minor axis (0, ±2)
Here, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form x2/a2 + y2/b2 = 1, where a is the
semi-major axis.
Accordingly, a = 3 and b = 2.
Thus, the equation of the ellipse is x2/32 + y2/22 = 1 or x2/9 + y2/4 = 1
Question 14:
Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0, ±√5), ends of minor axis (±1, 0)
Answer:
Given, Ends of major axis (0, ±√5), ends of minor axis (±1, 0)
Here, the major axis is along the y-axis.
Therefore, the equation of the ellipse will be of the form x2/b2 + y2/a2 = 1, where a is the
semi-major axis.
Accordingly, a = √5 and b = 1.
Thus, the equation of the ellipse is x2/12 + y2/(√5)2 = 1 or x2/1 + y2/5 = 1
Question 15:
Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0).
Answer:
Length of major axis = 26; foci = (±5, 0).
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form x2/a2 + y2/b2 = 1, where a is the
semi-major axis.
Accordingly, 2a = 26
=> a = 13 and c = 5.
We know that
a2 = b2 + c2
=> 132 = b2 + 52
=> b2 = 169 - 25
=> b2 = 144
=> b = 12
Thus, the equation of the ellipse is x2/132 + y2/122 = 1 or x2/169 + y2/144 = 1
Question 16:
Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6).
Answer:
Length of minor axis = 16, foci = (0, ±6).
Since the foci are on the y-axis, the major axis is along the y-axis.
Therefore, the equation of the ellipse will be of the form x2/b2 + y2/a2 = 1, where a is the
semi-major axis.
Accordingly, 2b = 16
=> b = 8 and c = 6.
We know that
a2 = b2 + c2
=> a2 = 82 + 62
=> a2 = 64 + 36
=> a2 = 100
=> a = 10
Thus, the equation of the ellipse is x2/82 + y2/102 = 1 or x2/64 + y2/100 = 1
Question 17:
Find the equation for the ellipse that satisfies the given conditions: foci (±3, 0), a = 4
Answer:
Given, foci (±3, 0), a = 4.
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form x2/a2 + y2/b2 = 1, where a is the
semi-major axis.
Accordingly, a = 4 and c = 3.
We know that
a2 = b2 + c2
=> 42 = b2 + 32
=> b2 = 16 - 9
=> b2 = 7
Thus, the equation of the ellipse is x2/16 + y2/7 = 1
Question 18:
Find the equation for the ellipse that satisfies the given conditions: b = 3, c = 4, centre at the origin; foci on the x axis.
Answer:
It is given that b = 3, c = 4, centre at the origin; foci on the x axis.
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form x2/a2 + y2/b2 = 1, where a is the
semi-major axis.
Accordingly, b = 3, c = 4.
We know that
a2 = b2 + c2
=> a2 = 32 + 42
=> a2 = 9 + 16
=> a2 = 25
=> a = 5
Thus, the equation of the ellipse is x2/52 + y2/32 = 1 or x2/25 + y2/9 = 1
Question 19:
Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
Answer:
Since the centre is at (0, 0) and the major axis is on the y-axis, the equation of the ellipse will
be of the form
x2/b2 + y2/a2 = 1 ………….1
where a is the semi-major axis.
The ellipse passes through points (3, 2) and (1, 6). Hence,
9/b2 + 4/a2 = 1 ………….2
1/b2 + 36/a2 = 1 ………….3
On solving equations 2 and 3, we get
b2 = 10 and a2 = 40
Thus, the equation of the ellipse is x2/10 + y2/40 = 1 or 4x2 + y2 = 40
Question 20:
Find the equation for the ellipse that satisfies the given conditions: Major axis on the x-axis and passes through the points (4, 3) and (6, 2).
Answer:
Since the major axis is on the x-axis, the equation of the ellipse will be of the form
x2/b2 + y2/a2 = 1 ………….1
where a is the semi-major axis.
The ellipse passes through points (4, 3) and (6, 2). Hence,
16/b2 + 9/a2 = 1 ………….2
36/b2 + 4/a2 = 1 ………….3
On solving equations 2 and 3, we get
a2 = 52 and b2 = 13.
Thus, the equation of the ellipse is x2/52 + y2/13 = 1 or x2 + 4y2 = 52
Exercise 11.4
Question 1:
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola x2/16 - y2/9 = 1.
Answer:
The given equation is x2/16 - y2/9 = 1
=> x2/42 - y2/32 = 1
On comparing this equation with the standard equation of hyperbola x2/a2 - y2/b2 = 1, we get
a = 4 and b = 3
We know that a2 + b2 = c2
=> 42 + 32 = c2
=> c2 = 16 + 9
=> c2 = 25
=> c = 5
Therefore,
The coordinates of the foci are (±c, 0) i.e. (±5, 0).
The coordinates of the vertices are (±a, 0) i.e (±4, 0).
Eccentricity e = c/a = 5/4
Length of latus rectum = 2b2/a = (2 * 9)/4 = 9/2
Question 2:
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola y2/9 - x2/27 = 1.
Answer:
The given equation is y2/9 - x2/27 = 1
=> y2/32 - x2/(√27)2 = 1
On comparing this equation with the standard equation of hyperbola y2/a2 - x2/b2 = 1, we get
a = 3 and b = √27
We know that a2 + b2 = c2
=> 32 + (√27)2 = c2
=> c2 = 9 + 27
=> c2 = 36
=> c = 6
Therefore,
The coordinates of the foci are (0, ±c) i.e. (0, ±6).
The coordinates of the vertices are (0, ±a) i.e (0, ±3).
Eccentricity, e = c/a = 6/3 = 2
Length of latus rectum = 2b2/a = (2 * 27)/3 = 2 * 9 = 18
Question 3:
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 9y2 - 4x2 = 36.
Answer:
The given equation is 9y2 - 4x2 = 36
=> 9y2/36 - 4x2/36 = 1
=> y2/4 - x2/9 = 1
=> y2/22 - x2/32 = 1 ……………..1
On comparing this equation with the standard equation of hyperbola y2/a2 - x2/b2 = 1, we get
a = 2 and b = 3
We know that a2 + b2 = c2
=> 22 + 32 = c2
=> c2 = 4 + 9
=> c2 = 13
=> c = √13
Therefore,
The coordinates of the foci are (0, ±c) i.e. (0, ±√13).
The coordinates of the vertices are (0, ±a) i.e (0, ±2).
Eccentricity, e = c/a = √13/2
Length of latus rectum = 2b2/a = (2 * 9)/2 = 9
Question 4:
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 16x2 – 9y2 = 576.
Answer:
The given equation is 16x2 – 9y2 = 576
=> 16x2/576 – 9y2/576 = 1
=> x2/36 - y2/64 = 1
=> x2/62 - y2/82 = 1 ……………..1
On comparing this equation with the standard equation of hyperbola x2/a2 - y2/b2 = 1, we get
a = 6 and b = 8
We know that a2 + b2 = c2
=> 62 + 82 = c2
=> c2 = 36 + 64
=> c2 = 100
=> c = 10
Therefore,
The coordinates of the foci are (±c, 0) i.e. (±10, 0).
The coordinates of the vertices are (±a, 0) i.e (±6, 0).
Eccentricity e = c/a = 10/6 = 5/3
Length of latus rectum = 2b2/a = (2 * 64)/6 = 64/3
Question 5:
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y2 - 9x2 = 36.
Answer:
The given equation is 5y2 - 9x2 = 36
=> 5y2/36 - 9x2/36 = 1
=> y2/(36/5) - x2/4 = 1
=> y2/(6/√5)2 - x2/22 = 1 ……………..1
On comparing this equation with the standard equation of hyperbola y2/a2 - x2/b2 = 1, we get
a = 6/√5 and b = 2
We know that a2 + b2 = c2
=> (6/√5)2 + 22 = c2
=> c2 = 36/5 + 4
=> c2 = 56/5
=> c = √(56/5)
=> c = 2√(14/5)
Therefore,
The coordinates of the foci are (0, ±c) i.e. (0, ±2√(14/5)).
The coordinates of the vertices are (0, ±a) i.e (0, ±6/√5).
Eccentricity, e = {2√(14/5)}/(6/√5) = √14/3
Length of latus rectum = 2b2/a = (2 * 49)/( 6/√5)) = 4√5/3
Question 6:
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 49y2 - 16x2 = 784.
Answer:
The given equation is 49y2 - 16x2 = 784
=> 49y2/784 - 16x2/784 = 1
=> y2/16 - x2/49 = 1
=> y2/42 - x2/72 = 1 ……………..1
On comparing this equation with the standard equation of hyperbola y2/a2 - x2/b2 = 1, we get
a = 4 and b = 7
We know that a2 + b2 = c2
=> 42 + 72 = c2
=> c2 = 16 + 49
=> c2 = 65
=> c = √65
Therefore,
The coordinates of the foci are (0, ±c) i.e. (0, ±√65).
The coordinates of the vertices are (0, ±a) i.e (0, ±4).
Eccentricity, e = c/a = √65/4
Length of latus rectum = 2b2/a = (2 * 49)/4 = 49/2
Question 7:
Find the equation of the hyperbola satisfying the give conditions: Vertices (±2, 0), foci (±3, 0).
Answer:
Given, Vertices (±2, 0), foci (±3, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form x2/a2 - y2/b2 = 1
Since the vertices are (±2, 0), a = 2
Since the foci are (±3, 0), c = 3
We know that a2 + b2 = c2
=> 22 + b2 = 32
=> b2 = 9 - 4
=> b2 = 5
Thus, the equation of the hyperbola is x2/4 - y2/5 = 1
Question 8:
Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±5), foci (0, ±8).
Answer:
Given, Vertices (0, ±5), foci (0, ±8)
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form y2/a2 - x2/b2 = 1
Since the vertices are (0, ±5), a = 5
Since the foci are (0, ±8), c = 8
We know that a2 + b2 = c2
=> 52 + b2 = 82
=> b2 = 64 - 25
=> b2 = 39
Thus, the equation of the hyperbola is y2/25 - x2/39 = 1
Question 9:
Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±3), foci (0, ±5).
Answer:
Given, Vertices (0, ±3), foci (0, ±5)
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form y2/a2 - x2/b2 = 1
Since the vertices are (0, ±3), a = 3
Since the foci are (0, ±5), c = 5
We know that a2 + b2 = c2
=> 32 + b2 = 52
=> b2 = 25 - 9
=> b2 = 16
Thus, the equation of the hyperbola is y2/9 - x2/16 = 1
Question 10:
Find the equation of the hyperbola satisfying the give conditions: foci (±5, 0), the transverse axis is of length 8.
Answer:
Given, foci (±5, 0), the transverse axis is of length 8
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form x2/a2 - y2/b2 = 1
Since the foci are (±5, 0), c = 5
Since the length of transverse axis is 8
So, 2a = 8
=> a = 4
We know that a2 + b2 = c2
=> 42 + b2 = 52
=> b2 = 25 - 16
=> b2 = 9
Thus, the equation of the hyperbola is x2/16 - y2/9 = 1
Question 11:
Find the equation of the hyperbola satisfying the give conditions: foci (0, ±13), the conjugate axis is of length 24.
Answer:
Given, foci (0, ±13), the conjugate axis is of length 24.
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form y2/a2 - x2/b2 = 1
Since the foci are (0, ±13), c = 13
Since the length of conjugate axis is 24
So, 2b = 24
=> b = 12
We know that a2 + b2 = c2
=> a2 + 122 = 132
=> a2 = 169 - 144
=> a2 = 25
Thus, the equation of the hyperbola is y2/25 - x2/144 = 1
Question 12:
Find the equation of the hyperbola satisfying the give conditions: Foci (±3√5, 0), the latus rectum is of length 8.
Answer:
Given, Foci = (±3√5, 0) and the latus rectum is of length 8.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form x2/a2 - y2/b2 = 1.
Since the foci are (±3√5, 0), c = ±3√5
Length of latus rectum = 8
=> 2b2/a = 8
=> b2/a = 4
=> b2 = 4a
We know that a2 + b2 = c2
=> a2 + 4a = (±3√5)2
=> a2 + 4a = 45
=> a2 + 4a – 45 = 0
=> (a + 9)(a - 5) = 0
=> a = -9, 5
Since a is non-negative, a = 5.
So, b2 = 4a = 4 * 5 = 20
Thus, the equation of the hyperbola is x2/25 - y2/20 = 1
Question 13:
Find the equation of the hyperbola satisfying the give conditions: Foci (±4, 0), the latus rectum is of length 12.
Answer:
Given, Foci = (±4, 0) and the latus rectum is of length 12.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form x2/a2 - y2/b2 = 1.
Since the foci are (±4, 0), c = 4
Length of latus rectum = 12
=> 2b2/a = 12
=> b2/a = 6
=> b2 = 6a
We know that a2 + b2 = c2
=> a2 + 6a = 42
=> a2 + 6a = 16
=> a2 + 6a – 16 = 0
=> (a + 8)(a - 2) = 0
=> a = -8, 2
Since a is non-negative, a = 2
So, b2 = 6a = 6 * 2 = 12
Thus, the equation of the hyperbola is x2/4 - y2/12 = 1
Question 14:
Find the equation of the hyperbola satisfying the give conditions: Vertices (±7, 0), e = 4/3.
Answer:
Given, Vertices (±7, 0), e = 4/3
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form x2/a2 - y2/b2 = 1.
Since the vertices are (±7, 0), a = 7
It is given that e = 4/3
So, c/a = 4/3
=> c/7 = 4/3
=> c = 28/3
We know that a2 + b2 = c2
=> 72 + b2 = (28/3)2
=> b2 = 784/9 – 49
=> b2 = (789 – 441)/9
=> a2 = 343/9
Thus, the equation of the hyperbola is x2/49 - y2/(343/9) = 1
=> x2/49 - 9y2/343 = 1
Question 15:
Find the equation of the hyperbola satisfying the give conditions: Foci (0, ±√10), passing through (2, 3).
Answer:
Given, Foci (0, ±√10), passing through (2, 3).
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form y2/a2 - x2/b2 = 1.
Since the foci are (0, ±√10), c = √10
We know that a2 + b2 = c2
=> a2 + b2 = 10
=> b2 = 10 - a2 ……………1
Since the hyperbola passes through point (2, 3),
9/a2 - 4/b2 = 1 ………..2
From equations 1 and 2, we obtain
9/a2 - 4/(10 - a2)= 1
=> 9(10 - a2) - 4a2 = a2 (10 - a2)
=> 900 - 9a2 - 4a2 = 10a2 – a4
=> a4 – 23a2 + 90 = 0
=> a4 – 18a2 - 5a2 + 90 = 0
=> a2 (a2 - 18) - 5(a2 - 18) = 0
=> (a2 - 18)(a2 - 5) = 0
=> a2 = 18, 5
In hyperbola, c > a,
=> c2 > a2
So, a2 = 5
Now, b2 = 10 – a2
=> b2 = 10 – 5
=> b2 = 5
Thus, the equation of the hyperbola is y2/5 - x2/5 = 1.
Miscellaneous Exercise 11
Question 1:
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
Answer:
The origin of the coordinate plane is taken at the vertex of the parabolic reflector in such a
way that the axis of the reflector is along the positive x-axis.
This can be diagrammatically represented as
The equation of the parabola is of the form y2 = 4ax (as it is opening to the right).
Since the parabola passes through point A(10, 5),
So, 102 = 4a * 5
=> 100 = 20a
=> a = 100/20
=> a = 5
Therefore, the focus of the parabola is (a, 0) = (5, 0), which is the mid-point of the diameter.
Hence, the focus of the reflector is at the mid-point of the diameter.
Question 2:
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base.
How wide is it 2 m from the vertex of the parabola?
Answer:
The origin of the coordinate plane is taken at the vertex of the arch in such a way that its
vertical axis is along the negative y-axis.
This can be diagrammatically represented as
The equation of the parabola is of the form x2 = -4ay (as it is opening downwards).
Given, width of parabola = AB = 5 m
So, BC = 5/2 cm [Since parabola is symmetric about its axis]
Again, parabola is 10 m high
So, OC = 10 m
=> BD = OC = 10 m
Hence, the coordinate of the point B is (5/2, -10).
Again point B lies on the parabola
So, (5/2)2 = -4a * (-10)
=> 25/4 = 40a
=> 4 * 40a = 25
=> 160a = 25
=> a = 25/160
=> a = 5/32
So, the arch is in the form of a parabola whose equation is
x2 = -4 * (5/32) * y
=> x2 = -5y/8
Let point P be point 2 m from the vertex of parabola.
So, OP = 2 m
Now, when y = 2,
x2 = 4 * (5/32) * 2
=> x2 = 5/4
=> x = √(5/4)
=> PN = √(5/4)
Now, MN = 2PN = 2 * √(5/4) = 2 * √5/2 = √5 = 2.23 (approx.)
Hence, when the arch is 2 m from the vertex of the parabola, its width is approximately
2.23 m.
Question 3:
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.
The road way which is horizontal and 100 m long is supported by vertical wires attached to the cable,
the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
Answer:
The vertex is at the lowest point of the cable. The origin of the coordinate plane is taken as the
vertex of the parabola, while its vertical axis is taken along the positive y-axis. This can be
diagrammatically represented as
Here, AB and OC are the longest and the shortest wires, respectively, attached to the cable.
DF is the supporting wire attached to the roadway, 18 m from the middle.
Here, AB = 30 m, OC = 6 m, and BC = 100/2 = 50 m.
The equation of the parabola is of the form x2 = 4ay (as it is opening upwards).
The coordinates of point A are (50, 30 – 6) = (50, 24).
Since A (50, 24) is a point on the parabola,
So, 502 = 4a * 24
=> 2500 = 96a
=> a = 2500/96
=> a = 625/24
Now, equation of the parabola is
x2 = 4 * (625/24) * y
=> x2 = 625y/6
=> 6x2 = 625y
The x-coordinate of point D is 18.
Hence, at x = 18,
6 * 182 = 625y
=> 1944 = 625y
=> y = 1944/625
=> y = 3.11 (approx.)
So, DE = 3.11 m
Now, DF = DE + EF = 3.11 m + 6 m = 9.11 m
Thus, the length of the supporting wire attached to the roadway 18 m from the middle is
approximately 9.11 m.
Question 4:
An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.
Find the height of the arch at a point 1.5 m from one end.
Answer:
Since the height and width of the arc from the centre is 2 m and 8 m respectively, it is clear
that the length of the major axis is 8 m, while the length of the semi-minor axis is 2 m.
The origin of the coordinate plane is taken as the centre of the ellipse, while the major axis is
taken along the x-axis. Hence, the semi-ellipse can be diagrammatically represented as
The equation of the semi-ellipse will be of the form x2/a2 + y2/b2 = 1, y ≥ 0 where a is the
semi-major axis
Accordingly, 2a = 8
=> a = 4
and b = 2
Therefore, the equation of the semi-ellipse is x2/16 + y2/4 = 1, y ≥ 0 …………..1
Let A be a point on the major axis such that AB = 1.5 m.
Draw AC⊥ OB.
OA = (4 – 1.5) m = 2.5 m
The x-coordinate of point C is 2.5.
On substituting the value of x with 2.5 in equation 1, we get
(2.5)2/16 + y2/4 = 1
=> 6.25/16 + y2/4 = 1
=> y2/4 = 1 – 6.25/16
=> y2/4 = (16 – 6.25)/16
=> y2/4 = 9.75/16
=> y2 = (4 *9.75)/16
=> y2 = 9.75/4
=> y2 = 2.4375
=> y = √2.4375
=> y = 1.56 (approx.)
So, AC = 1.56 m
Thus, the height of the arch at a point 1.5 m from one end is approximately 1.56 m.
Question 5:
A rod of length 12 cm moves with its ends always touching the coordinate axes.
Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.
Answer:
Let AB be the rod making an angle θ with OX and P (x, y) be the point on it such that AP = 3 cm.
Then, PB = AB – AP = (12 – 3) cm = 9 cm [AB = 12 cm]
From P, draw PQ⊥OY and PR ⊥ OX.
In ΔPBQ, cos θ = PQ/PB = x/9
In ΔPRA, sin θ = PR/PA = y/3
We know that,
sin2 θ + cos2 θ = 1
=> (y/3)2 + (x/9)2 = 1
=> x2/81 + y2/9 = 1
Thus, the equation of the locus of point P on the rod is x2/81 + y2/9 = 1.
Question 6:
Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum.
Answer:
The given parabola is x2 = 12y.
On comparing this equation with x2 = 4ay, we obtain
4a = 12
=> a = 3
So, the coordinates of foci are S(0, a) = S(0, 3)
Let AB be the latus rectum of the given parabola.
The given parabola can be roughly drawn as
At y = 3,
x2 = 12 * 3
=> x2 = 36
=> x = ±6
So, the coordinates of A is (–6, 3) and the coordinates of B is (6, 3).
Therefore, the vertices of ∆OAB are O(0, 0), A(–6, 3), and B(6, 3).
Area of ΔOAB = (1/2) * |0(3 - 3) + (-6)(3 - 0) + 6(0 - 3)|
= (1/2) * |-18 - 18|
= (1/2) * |-36|
= 36/2
= 18
Thus, the required area of the triangle is 18 unit2.
Question 7:
A man running a racecourse notes that the sum of the distances from the two flag posts form
him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.
Answer:
Let A and B be the positions of the two flag posts and P(x, y) be the position of the man.
Let the equation of the ellipse be: x2/a2 + y2 /b2 = 1
It is given that the sum of the distances of the man from the two flag posts is 10 m.
This means that the sum of focal distances of a point on the ellipse is 10 m
=> PA + PB = 10
=> 2a = 10
=> a = 5
Again, given that the distance between the flag posts is 8 meters
=> 2ae = 8
=> ae = 4
Now, b2 = a2 (1 - e2)
=> b2 = a2 - a2e2
=> b2 = a2 - (ae)2
=> b2 = 52 - 42
=> b2 = 25 - 16
=> b2 = 9
=> b = 3
Hence, the equation of the path is x2/52 + y2/32 = 1
=> x2/25 + y2/9 = 1
Question 8:
An equilateral triangle is inscribed in the parabola y2 = 4ax, where one vertex is at the vertex of the parabola.
Find the length of the side of the triangle.
Answer:
Let OAB be the equilateral triangle inscribed in parabola y2 = 4ax.
Let AB intersect the x-axis at point C.
Let OC = k
From the equation of the given parabola, we have
y2 = 4ak
=> y = ±2√(ak)
So, the respective coordinates of points A and B are (k, 2√(ak)) and (k, -2√(ak))
Now, AB = CA + CB = 2√(ak) + 2√(ak) = 4√(ak)
Since OAB is an equilateral triangle,
OA2 = AB2
=> k2 + {2√(ak)}2 = {4√(ak)}2
=> k2 + 4ak = 16ak
=> k2 = 16ak – 4ak
=> k2 = 12ak
=> k = 12a
Now, AB = 4√(ak)
=> AB = 4√(a * 12a)
=> AB = 4 * a * 2√3
=> AB = 8a√3
Thus, the side of the equilateral triangle inscribed in parabola y2 = 4 ax is 8a√3 unit.
Ncert Solutions Conic Sections Class 11
Source: https://www.examfear.com/cbse-ncert-solution/Class-11/Maths/Conic-Sections/solutions.htm
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