# NCERT Solutions for Class 11 Math Chapter 11 - Conic Sections

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The topics included in the chapter “Conic Sections” of Class 11 Maths are standard equations, ellipse and hyperbola, sections of a cone: hyperbola, straight line, circle, parabola, a point, ellipse, and pair of intersecting lines as a degenerated case of a conic section; standard equation of a circle, simple properties of parabola, etc.

##### Question 1:

Find an equation of the circle with centre at (0, 0) and radius r.

##### Question 2:

Find the equation of the circle with centre (–3, 2) and radius 4.

##### Question 4:

Find the equation of the circle which passes through the points (2, –2) and (3, 4) and whose centre lies on the line x + y = 2.

##### Question 5:

Find the co-ordinates of the focus, axis, the equation of the directrix and latus-rectum of the parabola y2 = 8x.

The given parabola is y2 = 8x ...(1)
Comparing with y2 = 4ax, 4a = 8 $⇒$ a = 2.
(i) Focus is (a, 0) i.e.(2, 0).
(ii) Equation of directrix is x = – a
i.e. x = – 2.
(iii) Length of latus-rectum = 4a = 4(2) = 8.

##### Question 6:

Find the equation of the parabola with focus (2, 0) and directrix x = – 2.

##### Question 7:

Find the equation of the parabola with vertex at (0, 0) and focus is (0, 2).

##### Question 8:

Find the equation of the parabola which is symmetric about y-axis and passes through the point (2, – 3).

##### Question 10:

(III) Vertices are (0, $±$ a) i.e. (0, $±$ 3). (IV) Length of major axis = 2a = 2(3) = 6. (V) Length of minor axis = 2b = 2(2) = 4.

##### Question 11:

Find the equation of the ellipse whose vertices are ($±$ 13, 0) and foci are ($±$ 5, 0).

##### Question 12:

Find the equation of the ellipse, whose length of major axis is 20 and foci are (0, $±$ 5).

##### Question 13:

Find the equation of the ellipse, with major axis along the x-axis and passing through the points (4, 3) and (–1, 4).

##### Question 14:

Find the co-ordinates of the foci and the vertices, the eccentricity, the length of the latus-rectum of the hyperbolas :

##### Question 15:

Find the co-ordinates of the foci and the vertices, the eccentricity, the length of the latus-rectum of the hyperbolas :

##### Question 17:

Find the equation of the hyperbola whose foci are (0, $±$ 12) and length of the latus-rectum is 36.

##### Question 18:

The focus of a parabolic mirror as shown in the figure is at a distance of 5 cm from its vertex. If the mirror is 45 cm deep, find the distance AB.

Here OS = Distance of focus from vertex = 5 cm.
Thus a = 5.

##### Question 19:

A beam is supported at its end points by supports which are 12 metres apart. Since the load is concentrated at its centre, there is a deflection of 3 cm at the centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection 1 cm ?

##### Question 20:

A rod AB of length 15 cm rests in between two co-ordinate axes in such a way that the end point A lies on the x-axis and end point B lies on y-axis. A point P (x, y) is taken on the rod in such a way that AP = 6 cm. Show that the locus of P is an ellipse.

##### Question 21:

In each of the following, find the equation of the circle with :
Centre (0, 2) and radius 2.

##### Question 22:

In each of the following, find the equation of the circle with :
Centre (– 2, 3) and radius 4.

##### Question 23:

In each of the following, find the equation of the circle with :

##### Question 24:

In each of the following, find the equation of the circle with :

##### Question 25:

In each of the following, find the equation of the circle with :

##### Question 26:

In each of the following , find the centre and radius of the circles :

##### Question 27:

In each of the following , find the centre and radius of the circles :

##### Question 28:

In each of the following , find the centre and radius of the circles :

##### Question 29:

In each of the following , find the centre and radius of the circles :

##### Question 30:

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.

Let the equation of the circle be :
x2 + y2 + 2gx + 2fy + c = 0 ...(1)
Since (1) passes thro’ (4, 1),
∴16 + 1 + 2g (4) + 2f (1) + c = 0
$⇒$ 17 + 8g + 2f + c = 0 ...(2)
Since (1) passes thro’ (6, 5),
∴36 + 25 + 2g (6) + 2f (5) + c = 0
$⇒$ 61 + 12g + 10f + c = 0 ...(3)
Since the centre (– g, – f) lies on 4x + y = 16,
∴ 4 (– g) + (– f) = 16
$⇒$ 4g + f = – 16 ...(4)
Subtracting (2) from (3),
44 + 4g + 8f = 0
$⇒$ 4g + 8f = – 44 ...(5)
Subtracting (4) from (5),
7f = – 28
$⇒$ f = – 4.
Putting in (4),
4g – 4 = – 16
$⇒$ 4g = – 12
g = – 3.
Putting in (2)
17 + 8 (– 3) + 2 (– 4) + c = 0
$⇒$ 17 – 24 – 8 + c =0

$⇒$ c = 15.
Putting the values of g, f and c in (1), we get :
x2 + y2 – 6x – 8y + 15 = 0,
which is the reqd. equation.

##### Question 31:

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.

##### Question 32:

Find the equation of the circle of radius 5 whose centre lies on x-axis and passes through the point (2, 3).

##### Question 33:

Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.

##### Question 34:

Find the equation of the circle whose centre is (2, 2) and which passes through the point (4, 5).

##### Question 36:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
y2 = 12x.

The given parabola is y2 = 12x.
Here 4a = 12
$⇒$ a = 3.
(i) Focus is (a, 0), i.e. (3, 0).
(ii) Axis of parabola is x-axis.
(iii) Equation of directrix is x = – a,
i.e. x = – 3.
(iv) Length of latus rectum = 4a = 4 (3) = 12.

##### Question 37:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum x2 = 6y.

##### Question 38:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
y2 = – 8x.

The given parabola is y2 = – 8x.
Here 4a = 8 $⇒$ a = 2.
(i) Focus is (– a, 0) i.e. (– 2, 0).
(ii) Axis of parabola is x-axis.
(iii) Equation of directrix is x = a i.e. x = 2.
(iv) Length latus-rectum = 4a = 4 (2) = 8.

##### Question 39:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
x2 = – 16y.

The given parabola is x2 = – 16y
Here 4a = 16 $⇒$ a = 4.
(i) Focus is (0, – a)i.e. (0, – 4).
(ii) Axis of parabola is y-axis.
(iii) Equation of directrix is y = a i.e. y = 4.
(iv) Length of latus-rectum = 4a = 4 (4) = 16.

##### Question 40:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
y2 = 10x.

##### Question 41:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
x2 = – 9y.

##### Question 42:

Find the equation of the parabola that satisfy the given conditions :
Focus (6, 0), directrix x = – 6.

##### Question 43:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
y2 = – 8x.

The given parabola is y2 = – 8x.
Here 4a = 8 $⇒$ a = 2.
(i) Focus is (– a, 0) i.e. (– 2, 0).
(ii) Axis of parabola is x-axis.
(iii) Equation of directrix is x = a i.e. x = 2.
(iv) Length latus-rectum = 4a = 4 (2) = 8.

##### Question 44:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
x2 = – 16y.

The given parabola is x2 = – 16y
Here 4a = 16 $⇒$ a = 4.
(i) Focus is (0, – a)i.e. (0, – 4).
(ii) Axis of parabola is y-axis.
(iii) Equation of directrix is y = a i.e. y = 4.
(iv) Length of latus-rectum = 4a = 4 (4) = 16.

##### Question 45:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
y2 = 10x.

##### Question 46:

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and length of latus rectum :
x2 = – 9y.

##### Question 47:

Find the equation of the parabola that satisfy the given conditions :
Focus (6, 0), directrix x = – 6.

##### Question 48:

Find the equation of the parabola that satisfy the given conditions :
Focus (0, – 3), directrix y = 3.

##### Question 49:

Find the equation of the parabola that satisfy the given conditions :
Vertex (0, 0), focus (3, 0).

##### Question 50:

Find the equation of the parabola that satisfy the given conditions :
Vertex (0, 0), focus (– 2, 0).

##### Question 51:

Find the equation of the parabola that satisfy the given conditions :
Vertox (0, 0), passing through (2, 3) and axis is along x-axis.

##### Question 52:

Find the equation of the parabola that satisfy the given conditions :
Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis.

##### Question 53:

Find the co-ordinates 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 :

##### Question 54:

Find the co-ordinates 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 :

##### Question 55:

Find the co-ordinates 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 :

##### Question 56:

Find the co-ordinates 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 :

##### Question 57:

Find the co-ordinates 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 :

##### Question 58:

Find the co-ordinates 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 :

##### Question 59:

Find the co-ordinates 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.

##### Question 60:

Find the co-ordinates 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.

##### Question 61:

Find the co-ordinates 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.

##### Question 62:

Find the equation for the ellipse that satisfies the given conditions :
Vertices ($±$ 5, 0); foci ($±$ 4, 0).

##### Question 63:

Find the equation for the ellipse that satisfies the given conditions :
Vertices (0, $±$ 13); foci (0, $±$ 5).

##### Question 64:

Find the equation for the ellipse that satisfies the given conditions :
Vertices ($±$ 6, 0); foci ($±$ 4, 0).

##### Question 65:

Find the equation for the ellipse that satisfies the given conditions :
Ends of major-axis ($±$ 3, 0); ends of minor axis (0, $±$ 2).

##### Question 66:

Find the equation for the ellipse that satisfies the given conditions :

##### Question 67:

Find the equation for the ellipse that satisfies the given conditions :
Length of major axis 26; foci ($±$ 5, 0).

##### Question 68:

Find the equation for the ellipse that satisfies the given conditions :
Length of minor-axis 16; foci (0, $±$ 6).

##### Question 69:

Find the equation for the ellipse that satisfies the given conditions :
Foci ($±$ 3, 0) ; a = 4.

##### Question 70:

Find the equation for the ellipse that satisfies the given conditions :
b = 3, c = 4, centre at the origin; focus on x–axis.

##### Question 71:

Find the equation for the ellipse that satisfies the given conditions :
Centre at (0, 0); major axis on the y–axis and passes through (3, 2) and (1, 6).

##### Question 72:

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).

##### Question 73:

Find the co-ordinates of the foci and the vertices, the eccentricity and length of the latus-rectum of the hyperbola :

##### Question 74:

Find the co-ordinates of the foci and the vertices, the eccentricity and length of the latus-rectum of the hyperbola :

##### Question 75:

Find the co-ordinates of the foci and the vertices, the eccentricity and length of the latus-rectum of the hyperbola :
9y2 – 4x2 = 36.

##### Question 76:

Find the co-ordinates of the foci and the vertices, the eccentricity and length of the latus-rectum of the hyperbola :
16x2 – 9y2 = 576.

##### Question 77:

Find the co-ordinates of the foci and the vertices, the eccentricity and length of the latus-rectum of the hyperbola :
5y2 – 9x2 = 36.

##### Question 78:

Find the co-ordinates of the foci and the vertices, the eccentricity and length of the latus-rectum of the hyperbola :
49y2 – 16x2 = 784.

##### Question 79:

Find the equation of the hyperbola satisfying the given conditions :
Vertices ($±$ 2, 0); foci ($±$ 3, 0).

##### Question 80:

Find the equation of the hyperbola satisfying the given conditions :
Vertices (0, $±$ 5) ; foci (0, $±$ 8).

##### Question 81:

Find the equation of the hyperbola satisfying the given conditions :
Vertices (0, $±$ 3) ; foci (0, $±$ 5).

##### Question 82:

Find the equation of the hyperbola satisfying the given conditions :
Foci ($±$ 5, 0); the transverse axis is of length 8.

##### Question 83:

Find the equation of the hyperbola satisfying the given conditions :
Foci (0, $±$ 13) ; the conjugate axis is of length 24.

##### Question 84:

Find the equation of the hyperbola satisfying the given conditions :

##### Question 85:

Find the equation of the hyperbola satisfying the given conditions :
Foci ($±$ 4, 0); the latus-rectum is of length 12.

##### Question 86:

Find the equation of the hyperbola satisfying the given conditions :

##### Question 87:

Find the equation of the hyperbola satisfying the given conditions :