# Coordinate Geometry - Conic Section Test 1

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:If a point $$(x,\;y) \equiv (\tan \theta + \sin \theta ,\;\tan \theta - \sin \theta )$$, then locus of (x, y) is

$${({x^2}y)^{2/3}} + {(x{y^2})^{2/3}} = 1$$

$${x^2} - {y^2} = 4xy$$

$${({x^2} - {y^2})^2} = 16xy$$

$${x^2} - {y^2} = 6xy$$

## Questions 2 of 50

Question:Equation$$\sqrt {{{(x - 2)}^2} + {y^2}} + \sqrt {{{(x + 2)}^2} + {y^2}} = 4$$represents

Parabola

Ellipse

Circle

Pair of straight lines

## Questions 3 of 50

Question:If the parabola $${y^2} = 4ax$$ passes through (-3, 2), then length of its latus rectum is

2/3

1/3

4/3

4

## Questions 4 of 50

Question:The ends of latus rectum of parabola $${x^2} + 8y = 0$$ are

(–4, –2) and (4, 2)

(4, –2) and (–4, 2)

(–4, –2) and (4, –2)

(4, 2) and (–4, 2)

## Questions 5 of 50

Question:The equation of the parabola whose vertex is  (-1, -2), axis is vertical and which passes through the point (3, 6), is

$${x^2} + 2x - 2y - 3 = 0$$

$$2{x^2} = 3y$$

$${x^2} - 2x - y + 3 = 0$$

None of these

## Questions 6 of 50

Question:Axis of the parabola $${x^2} - 4x - 3y + 10 = 0$$ is

$$y + 2 = 0$$

$$x + 2 = 0$$

$$y - 2 = 0$$

$$x - 2 = 0$$

## Questions 7 of 50

Question:If the vertex of the parabola $$y = {x^2} - 8x + c$$ lies on x-axis, then the value of c is

16

4

4

16

## Questions 8 of 50

Question:The points of intersection of the curves whose parametric equations are $$x = {t^2} + 1,\;y = 2t$$ and $$x = 2s,\;y = \frac{2}{s}$$ is given by

$$(1,\; - 3)$$

(2, 2)

(–2, 4)

(1, 2)

## Questions 9 of 50

Question:The vertex of parabola $${(y - 2)^2} = 16(x - 1)$$ is

(2, 1)

(1, –2)

(–1, 2)

(1, 2)

## Questions 10 of 50

Question:Equation of the parabola with its vertex at (1, 1) and focus (3, 1) is

$${(x - 1)^2} = 8(y - 1)$$

$${(y - 1)^2} = 8(x - 3)$$

$${(y - 1)^2} = 8(x - 1)$$

$${(x - 3)^2} = 8(y - 1)$$

## Questions 11 of 50

Question:The line $$x\cos \alpha + y\sin \alpha = p$$ will touch the parabola $${y^2} = 4a(x + a)$$, if

$$p\cos \alpha + a = 0$$

$$p\cos \alpha - a = 0$$

$$a\cos \alpha + p = 0$$

$$a\cos \alpha - p = 0$$

## Questions 12 of 50

Question:The equation of a tangent to the parabola $${y^2} = 4ax$$ making an angle $$\theta$$ with x-axis is

$$y = x\cot \theta + a\tan \theta$$

$$x = y\tan \theta + a\cot \theta$$

$$y = x\tan \theta + a\cot \theta$$

None of these

## Questions 13 of 50

Question:The angle between the tangents drawn from the origin to the parabola $${y^2} = 4a(x - a)$$ is

$${90^o}$$

$${30^o}$$

$${\tan ^{ - 1}}\frac{1}{2}$$

$${45^o}$$

## Questions 14 of 50

Question:If line $$x = my + k$$ touches the parabola $${x^2} = 4ay$$, then $$k =$$

$$\frac{a}{m}$$

am

$$a{m^2}$$

$$- a{m^2}$$

## Questions 15 of 50

Question:The equation of the common tangent touching the circle $${(x - 3)^2} + {y^2} = 9$$ and the parabola $${y^2} = 4x$$ above the x-axis, is

$$\sqrt 3 y = 3x + 1$$

$$\sqrt 3 y = - (x + 3)$$

$$\sqrt 3 y = x + 3$$

$$\sqrt 3 y = - (3x + 1)$$

## Questions 16 of 50

Question:The point at which the line $$y = mx + c$$ touches the parabola $${y^2} = 4ax$$ is

$$\left( {\frac{a}{{{m^2}}},\;\frac{{2a}}{m}} \right)$$

$$\left( {\frac{a}{{{m^2}}},\;\frac{{ - 2a}}{m}} \right)$$

$$\left( { - \frac{a}{{{m^2}}},\;\frac{{2a}}{m}} \right)$$

$$\left( { - \frac{a}{{{m^2}}},\; - \frac{{2a}}{m}} \right)$$

## Questions 17 of 50

Question:The equation of normal to the parabola at the point $$\left( {\frac{a}{{{m^2}}},\;\frac{{2a}}{m}} \right)$$,is

$$y = {m^2}x - 2mx - a{m^3}$$

$${m^3}y = {m^2}x - 2a{m^2} - a$$

$${m^3}y = 2a{m^2} - {m^2}x + a$$

None of these

## Questions 18 of 50

Question:If the line $$2x + y + k = 0$$ is normal to the parabola $${y^2} = - 8x$$, then the value of k will be

$$- 16$$

$$- 8$$

$$- 24$$

24

## Questions 19 of 50

Question:If the normal to  $${y^2} = 12x$$ at (3, 6) meets the parabola again in (27, -18) and the circle on the normal chord as diameter is

$${x^2} + {y^2} + 30x + 12y - 27 = 0$$

$${x^2} + {y^2} + 30x + 12y + 27 = 0$$

$${x^2} + {y^2} - 30x - 12y - 27 = 0$$

$${x^2} + {y^2} - 30x + 12y - 27 = 0$$

## Questions 20 of 50

Question:The length of the normal chord to the parabola $${y^2} = 4x$$, which subtends right angle at the vertex is

$$6\sqrt 3$$

$$3\sqrt 3$$

2

1

## Questions 21 of 50

Question:The ends of the latus rectum of the conic $${x^2} + 10x - 16y + 25 = 0$$ are

(3, –4), (13, 4)

(–3, –4), (13, –4)

(3, 4), (–13, 4)

(5, –8), (–5, 8)

## Questions 22 of 50

Question:Tangent to the parabola $$y = {x^2} + 6$$ at (1, 7) touches the circle $${x^2} + {y^2} + 16x + 12y + c = 0$$ at the point

(–6, –9)

(–13, –9)

(–6, –7)

(13, 7)

## Questions 23 of 50

Question:The eccentricity of an ellipse is 2/3, latus rectum is 5 and centre is (0, 0). The equation of the ellipse is

$$\frac{{{x^2}}}{{81}} + \frac{{{y^2}}}{{45}} = 1$$

$$\frac{{4{x^2}}}{{81}} + \frac{{4{y^2}}}{{45}} = 1$$

$$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1$$

$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$

## Questions 24 of 50

Question:The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is

$${x^2} + 2{y^2} = 100$$

$${x^2} + \sqrt 2 {y^2} = 10$$

$${x^2} - 2{y^2} = 100$$

None of these

## Questions 25 of 50

Question:The equation of the ellipse whose latus rectum is 8 and whose eccentricity is $$\frac{1}{{\sqrt 2 }}$$, referred to the principal axes of coordinates, is

$$\frac{{{x^2}}}{{18}} + \frac{{{y^2}}}{{32}} = 1$$

$$\frac{{{x^2}}}{8} + \frac{{{y^2}}}{9} = 1$$

$$\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{32}} = 1$$

$$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{24}} = 1$$

## Questions 26 of 50

Question:Eccentricity of the ellipse whose latus rectum is equal to the distance between two focus points, is

$$\frac{{\sqrt 5 + 1}}{2}$$

$$9{x^2} + 5{y^2} - 30y = 0$$

$$\frac{{\sqrt 5 }}{2}$$

$$\frac{{\sqrt 3 }}{2}$$

## Questions 27 of 50

Question:The locus of a variable point whose distance from (-2, 0) is $$\frac{2}{3}$$ times its distance from the line $$x = - \frac{9}{2}$$, is

Ellipse

Parabola

Hyperbola

None of these

## Questions 28 of 50

Question:If $$P \equiv (x,\;y)$$, $${F_1} \equiv (3,\;0)$$, $${F_2} \equiv ( - 3,\;0)$$ and $$16{x^2} + 25{y^2} = 400$$, then $$P{F_1} + P{F_2}$$ equals

8

6

10

12

## Questions 29 of 50

Question:Latus rectum of ellipse $$4{x^2} + 9{y^2} - 8x - 36y + 4 = 0$$ is

8/3

4/3

$$\frac{{\sqrt 5 }}{3}$$

16/3

## Questions 30 of 50

Question:Eccentricity of the ellipse $$4{x^2} + {y^2} - 8x + 2y + 1 = 0$$ is

$$1/\sqrt 3$$

$$\sqrt 3 /2$$

$$1/2$$

None of these

## Questions 31 of 50

Question:The length of the axes of the conic $$9{x^2} + 4{y^2} - 6x + 4y + 1 = 0$$, are

$$\frac{1}{2},\;9$$

$$3,\;\frac{2}{5}$$

$$1,\;\frac{2}{3}$$

3, 2

## Questions 32 of 50

Question:The eccentricity of the ellipse $$9{x^2} + 5{y^2} - 18x - 2y - 16 = 0$$ is

1/2

2/3

1/3

3/4

## Questions 33 of 50

Question:The locus of the point of intersection of the perpendicular tangents to the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$$ is

$${x^2} + {y^2} = 9$$

$${x^2} + {y^2} = 4$$

$${x^2} + {y^2} = 13$$

$${x^2} + {y^2} = 5$$

## Questions 34 of 50

Question:The eccentric angles of the extremities of latus recta of the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ are given by

$${\tan ^{ - 1}}\left( { \pm \frac{{ae}}{b}} \right)$$

$${\tan ^{ - 1}}\left( { \pm \frac{{be}}{a}} \right)$$

$${\tan ^{ - 1}}\left( { \pm \frac{b}{{ae}}} \right)$$

$${\tan ^{ - 1}}\left( { \pm \frac{a}{{be}}} \right)$$

## Questions 35 of 50

Question:If the foci of an ellipse are $$( \pm \sqrt 5 ,\,0)$$ and its eccentricity is $$\frac{{\sqrt 5 }}{3}$$, then the equation of the ellipse is

$$9{x^2} + 4{y^2} = 36$$

$$4{x^2} + 9{y^2} = 36$$

$$36{x^2} + 9{y^2} = 4$$

$$9{x^2} + 36{y^2} = 4$$

## Questions 36 of 50

Question:The sum of the focal distances of any point on the conic $$\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1$$ is

10

9

41

18

## Questions 37 of 50

Question:The equation of the transverse and conjugate axis of the hyperbola $$16{x^2} - {y^2} + 64x + 4y + 44 = 0$$ are

$$x = 2,\;y + 2 = 0$$

$$x = 2,\;y = 2$$

$$y = 2,\;x + 2 = 0$$

None of these

## Questions 38 of 50

Question:If the length of the transverse and conjugate axes of a hyperbola be 8 and 6 respectively, then the difference focal distances of any point of the hyperbola will be

8

6

14

2

## Questions 39 of 50

Question:The length of transverse axis of the parabola $$3{x^2} - 4{y^2} = 32$$ is

$$\frac{{8\sqrt 2 }}{{\sqrt 3 }}$$

$$\frac{{16\sqrt 2 }}{{\sqrt 3 }}$$

$$\frac{3}{{32}}$$

$$\frac{{64}}{3}$$

## Questions 40 of 50

Question:The directrix of the hyperbola is $$\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4} = 1$$

$$x = 9/\sqrt {13}$$

$$y = 9/\sqrt {13}$$

$$x = 6/\sqrt {13}$$

$$y = 6/\sqrt {13}$$

## Questions 41 of 50

Question:The latus rectum of the hyperbola $$9{x^2} - 16{y^2} - 18x - 32y - 151 = 0$$ is

$$\frac{9}{4}$$

9

$$\frac{3}{2}$$

$$\frac{9}{2}$$

## Questions 42 of 50

Question:The equation of the hyperbola whose directrix is $$2x + y = 1$$, focus (1, 1) and eccentricity $$= \sqrt 3$$, is

$$7{x^2} + 12xy - 2{y^2} - 2x + 4y - 7 = 0$$

$$11{x^2} + 12xy + 2{y^2} - 10x - 4y + 1 = 0$$

$$11{x^2} + 12xy + 2{y^2} - 14x - 14y + 1 = 0$$

None of these

## Questions 43 of 50

Question:The equation of the tangent to the hyperbola $$4{y^2} = {x^2} - 1$$ at the point (1, 0) is

$$x = 1$$

$$y = 1$$

$$y = 4$$

$$x = 4$$

## Questions 44 of 50

Question:The value of m for which $$y = mx + 6$$ is a tangent to the hyperbola $$\frac{{{x^2}}}{{100}} - \frac{{{y^2}}}{{49}} = 1$$, is

$$\sqrt {\frac{{17}}{{20}}}$$

$$\sqrt {\frac{{20}}{{17}}}$$

$$\sqrt {\frac{3}{{20}}}$$

$$\sqrt {\frac{{20}}{3}}$$

## Questions 45 of 50

Question:The equation of the normal to the hyperbola $$\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$$ at the point $$(8,\;3\sqrt 3 )$$ is

$$\sqrt 3 x + 2y = 25$$

$$x + y = 25$$

$$y + 2x = 25$$

$$2x + \sqrt 3 y = 25$$

## Questions 46 of 50

Question:The equation of the normal at the point (6, 4) on the hyperbola $$\frac{{{x^2}}}{9} - \frac{{{y^2}}}{{16}} = 3$$, is

$$3x + 8y = 50$$

$$3x - 8y = 50$$

$$8x + 3y = 50$$

$$8x - 3y = 50$$

## Questions 47 of 50

Question:Curve $$xy = {c^2}$$ is said to be

Parabola

Rectangular hyperbola

Hyperbola

Ellipse

## Questions 48 of 50

Question:The reciprocal of the eccentricity of rectangular hyperbola, is

2

$$\frac{1}{2}$$

$$\sqrt 2$$

$$\frac{1}{{\sqrt 2 }}$$

## Questions 49 of 50

Question:If $$4{x^2} + p{y^2} = 45$$ and $${x^2} - 4{y^2} = 5$$ cut orthogonally, then the value of p is

1/9

1/3

3

18

9

## Questions 50 of 50

Question:Find the equation of axis of the given hyperbola $$\frac{{{x^2}}}{3} - \frac{{{y^2}}}{2} = 1$$ which is equally inclined to the axes

$$y = x + 1$$
$$y = x - 1$$
$$y = x + 2$$
$$y = x - 2$$