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

Answers Choices:

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

\({x^2} - {y^2} = 4xy\)

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

\({x^2} - {y^2} = 6xy\)

Questions 2 of 50

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

Answers Choices:

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

Answers Choices:

2/3

1/3

4/3

4

Questions 4 of 50

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

Answers Choices:

(–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

Answers Choices:

\({x^2} + 2x - 2y - 3 = 0\)

\(2{x^2} = 3y\)

\({x^2} - 2x - y + 3 = 0\)

None of these

Questions 6 of 50

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

Answers Choices:

\(y + 2 = 0\)

\(x + 2 = 0\)

\(y - 2 = 0\)

\(x - 2 = 0\)

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

Answers Choices:

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

Answers Choices:

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

(2, 2)

(–2, 4)

(1, 2)

Questions 9 of 50

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

Answers Choices:

(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

Answers Choices:

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

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

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

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

Questions 11 of 50

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

Answers Choices:

\(p\cos \alpha + a = 0\)

\(p\cos \alpha - a = 0\)

\(a\cos \alpha + p = 0\)

\(a\cos \alpha - p = 0\)

Questions 12 of 50

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

Answers Choices:

\(y = x\cot \theta + a\tan \theta \)

\(x = y\tan \theta + a\cot \theta \)

\(y = x\tan \theta + a\cot \theta \)

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

Answers Choices:

\({90^o}\)

\({30^o}\)

\({\tan ^{ - 1}}\frac{1}{2}\)

\({45^o}\)

Questions 14 of 50

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

Answers Choices:

\(\frac{a}{m}\)

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

Answers Choices:

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

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

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

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

Questions 16 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\(y = {m^2}x - 2mx - a{m^3}\)

\({m^3}y = {m^2}x - 2a{m^2} - a\)

\({m^3}y = 2a{m^2} - {m^2}x + a\)

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

Answers Choices:

\( - 16\)

\( - 8\)

\( - 24\)

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

Answers Choices:

\({x^2} + {y^2} + 30x + 12y - 27 = 0\)

\({x^2} + {y^2} + 30x + 12y + 27 = 0\)

\({x^2} + {y^2} - 30x - 12y - 27 = 0\)

\({x^2} + {y^2} - 30x + 12y - 27 = 0\)

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

Answers Choices:

\(6\sqrt 3 \)

\(3\sqrt 3 \)

2

1

Questions 21 of 50

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

Answers Choices:

(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

Answers Choices:

(–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

Answers Choices:

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

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

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

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

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

Answers Choices:

\({x^2} + 2{y^2} = 100\)

\({x^2} + \sqrt 2 {y^2} = 10\)

\({x^2} - 2{y^2} = 100\)

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

Answers Choices:

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

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

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

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

Questions 26 of 50

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

Answers Choices:

\(\frac{{\sqrt 5 + 1}}{2}\)

\(9{x^2} + 5{y^2} - 30y = 0\)

\(\frac{{\sqrt 5 }}{2}\)

\(\frac{{\sqrt 3 }}{2}\)

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

Answers Choices:

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

Answers Choices:

8

6

10

12

Questions 29 of 50

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

Answers Choices:

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

Answers Choices:

\(1/\sqrt 3 \)

\(\sqrt 3 /2\)

\(1/2\)

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

Answers Choices:

\(\frac{1}{2},\;9\)

\(3,\;\frac{2}{5}\)

\(1,\;\frac{2}{3}\)

3, 2

Questions 32 of 50

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

Answers Choices:

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

Answers Choices:

\({x^2} + {y^2} = 9\)

\({x^2} + {y^2} = 4\)

\({x^2} + {y^2} = 13\)

\({x^2} + {y^2} = 5\)

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\(9{x^2} + 4{y^2} = 36\)

\(4{x^2} + 9{y^2} = 36\)

\(36{x^2} + 9{y^2} = 4\)

\(9{x^2} + 36{y^2} = 4\)

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

Answers Choices:

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

Answers Choices:

\(x = 2,\;y + 2 = 0\)

\(x = 2,\;y = 2\)

\(y = 2,\;x + 2 = 0\)

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

Answers Choices:

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

Answers Choices:

\(\frac{{8\sqrt 2 }}{{\sqrt 3 }}\)

\(\frac{{16\sqrt 2 }}{{\sqrt 3 }}\)

\(\frac{3}{{32}}\)

\(\frac{{64}}{3}\)

Questions 40 of 50

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

Answers Choices:

\(x = 9/\sqrt {13} \)

\(y = 9/\sqrt {13} \)

\(x = 6/\sqrt {13} \)

\(y = 6/\sqrt {13} \)

Questions 41 of 50

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

Answers Choices:

\(\frac{9}{4}\)

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

Answers Choices:

\(7{x^2} + 12xy - 2{y^2} - 2x + 4y - 7 = 0\)

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

\(11{x^2} + 12xy + 2{y^2} - 14x - 14y + 1 = 0\)

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

Answers Choices:

\(x = 1\)

\(y = 1\)

\(y = 4\)

\(x = 4\)

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

Answers Choices:

\(\sqrt {\frac{{17}}{{20}}} \)

\(\sqrt {\frac{{20}}{{17}}} \)

\(\sqrt {\frac{3}{{20}}} \)

\(\sqrt {\frac{{20}}{3}} \)

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

Answers Choices:

\(\sqrt 3 x + 2y = 25\)

\(x + y = 25\)

\(y + 2x = 25\)

\(2x + \sqrt 3 y = 25\)

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

Answers Choices:

\(3x + 8y = 50\)

\(3x - 8y = 50\)

\(8x + 3y = 50\)

\(8x - 3y = 50\)

Questions 47 of 50

Question:Curve \(xy = {c^2}\) is said to be

Answers Choices:

Parabola

Rectangular hyperbola

Hyperbola

Ellipse

Questions 48 of 50

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

\(y = x + 1\)

\(y = x - 1\)

\(y = x + 2\)

\(y = x - 2\)