Coordinate Geometry - Conic Section Test 3

Total Questions:50 Total Time: 75 Min

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Questions 1 of 50

Question:PQ is a double ordinate of the parabola \({y^2} = 4ax\). The locus of the points of trisection of PQ is

Answers Choices:

\(9{y^2} = 4ax\)

\(9{x^2} = 4ay\)

\(9{y^2} + 4ax = 0\)

\(9{x^2} + 4ay = 0\)

Questions 2 of 50

Question:If the vertex of a parabola be at origin and directrix be \(x + 5 = 0\), then its latus rectum is

Answers Choices:

5

10

20

40

Questions 3 of 50

Question:The equation of the parabola with focus (3, 0) and the directirx \(x + 3 = 0\) is

Answers Choices:

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

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

\({y^2} = 12x\)

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

Questions 4 of 50

Question:Locus of the poles of focal chords of a parabola is of parabola

Answers Choices:

The tangent at the vertex

The axis

A focal chord

The directrix

Questions 5 of 50

Question:The equation of the parabola whose vertex and focus lies on the x-axis at distance a and a' from the origin, is

Answers Choices:

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

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

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

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

Questions 6 of 50

Question:The focus of the parabola \({y^2} = 4y - 4x\) is

Answers Choices:

(0, 2)

(1, 2)

(2, 0)

(2, 1)

Questions 7 of 50

Question:Vertex of the parabola \({x^2} + 4x + 2y - 7 = 0\) is

Answers Choices:

(–2, 11/2)

(–2, 2)

(–2, 11)

(2, 11)

Questions 8 of 50

Question:The focus of the parabola \({x^2} = 2x + 2y\) is

Answers Choices:

\(\left( {\frac{3}{2},\;\frac{{ - 1}}{2}} \right)\)

\(\left( {1,\;\frac{{ - 1}}{2}} \right)\)

(1, 0)

(0, 1)

Questions 9 of 50

Question:Latus rectum of the parabola \({y^2} - 4y - 2x - 8 = 0\) is

Answers Choices:

2

4

8

1

Questions 10 of 50

Question:The equation of the parabola with focus (a, b) and directrix \(\frac{x}{a} + \frac{y}{b} = 1\) is given by

Answers Choices:

\({(ax - by)^2} - 2{a^3}x - 2{b^3}y + {a^4} + {a^2}{b^2} + {b^4} = 0\)

\({(ax + by)^2} - 2{a^3}x - 2{b^3}y - {a^4} + {a^2}{b^2} - {b^4} = 0\)

\({(ax - by)^2} + {a^4} + {b^4} - 2{a^3}x = 0\)

\({(ax - by)^2} - 2{a^3}x = 0\)

Questions 11 of 50

Question:The equation of the parabola whose vertex is at (2, -1) and focus at (2, -3) is

Answers Choices:

\({x^2} + 4x - 8y - 12 = 0\)

\({x^2} - 4x + 8y + 12 = 0\)

\({x^2} + 8y = 12\)

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

Questions 12 of 50

Question:The directrix of the parabola \({x^2} - 4x - 8y + 12 = 0\) is

Answers Choices:

\(x = 1\)

\(y = 0\)

\(x = - 1\)

\(y = - 1\)

Questions 13 of 50

Question:The straight line \(y = 2x + \lambda \) does not meet the parabola \({y^2} = 2x\), if

Answers Choices:

\(\lambda < \frac{1}{4}\)

\(\lambda > \frac{1}{4}\)

\(\lambda = 4\)

\(\lambda = 1\)

Questions 14 of 50

Question:The equation of the tangent at a point \(P(t)\) where 't' is any parameter to the parabola \({y^2} = 4ax\), is

Answers Choices:

\(yt = x + a{t^2}\)

\(y = xt + a{t^2}\)

\(y = xt + \frac{a}{t}\)

\(y = tx\)

Questions 15 of 50

Question:The line \(y = 2x + c\) is tangent to the parabola \({y^2} = 4x\), then \(c = \)

Answers Choices:

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

\(\frac{1}{2}\)

\(\frac{1}{3}\)

4

Questions 16 of 50

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

Answers Choices:

\(a = c\)

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

\(m = {a^2}c\)

\(m = a{c^2}\)

Questions 17 of 50

Question:The angle of intersection between the curves \({x^2} = 4(y + 1)\) and \({x^2} = - 4(y + 1)\) is

Answers Choices:

\(\frac{\pi }{6}\)

\(\frac{\pi }{4}\)

0

\(\frac{\pi }{2}\)

Questions 18 of 50

Question:Angle between two curves \({y^2} = 4(x + 1)\) and \({x^2} = 4(y + 1)\) is

Answers Choices:

0 \(^{\rm{o}}\)

90 \(^{\rm{o}}\)

60 \(^{\rm{o}}\)

30 \(^{\rm{o}}\)

Questions 19 of 50

Question:The length of chord of contact of the tangents drawn from the point (2, 5) to the parabola \({y^2} = 8x\), is

Answers Choices:

\(\frac{1}{2}\sqrt {41} \)

\(\sqrt {41} \)

\(\frac{3}{2}\sqrt {41} \)

\(2\sqrt {41} \)

Questions 20 of 50

Question:If 'a' and 'c' are the segments of a focal chord of a parabola and b the semi-latus rectum, then

Answers Choices:

a, b, c are in A.P.

a, b, c are in G.P.

a, b, c are in H.P.

None of these

Questions 21 of 50

Question:The focal chord to \({y^2} = 16x\) is tangent to \({(x - 6)^2} + {y^2} = 2\), then the possible value of the slope of this chord, are

Answers Choices:

\(\{ - 1,\;1\} \)

{–2, 2}

{-2, 1/2}

{2, –1/2}

Questions 22 of 50

Question:The normal to the parabola \({y^2} = 8x\) at the point (2, 4) meets the parabola again at the point

Answers Choices:

{–18, –12}

{–18, 12}

{18, 12}

(18, –12)

Questions 23 of 50

Question:The equation to a parabola which passes through the intersection of a straight line \(x + y = 0\) and the circle \({x^2} + {y^2} + 4y = 0\) is

Answers Choices:

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

\({y^2} = x\)

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

None of these

Questions 24 of 50

Question:Let a circle tangent to the directrix of a parabola \({y^2} = 2ax\) has its centre coinciding with the focus of the parabola. Then the point of intersection of the parabola and circle is

Answers Choices:

(a, –a)

\((a/2,\;a/2)\)

\((a/2,\; \pm a)\)

\(( \pm a,\;a/2)\)

Questions 25 of 50

Question:The equation of the ellipse whose vertices are \(( \pm 5,\;0)\) and foci are \(( \pm 4,\;0)\) is

Answers Choices:

\(9{x^2} + 25{y^2} = 225\)

\(25{x^2} + 9{y^2} = 225\)

\(3{x^2} + 4{y^2} = 192\)

None of these

Questions 26 of 50

Question:The equation of the ellipse whose foci are \(( \pm 5,\;0)\) and one of its directrix is \(5x = 36\), is

Answers Choices:

\(\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{11}} = 1\)

\(\frac{{{x^2}}}{6} + \frac{{{y^2}}}{{\sqrt {11} }} = 1\)

\(\frac{{{x^2}}}{6} + \frac{{{y^2}}}{{11}} = 1\)

None of these

Questions 27 of 50

Question:If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is

Answers Choices:

\(\frac{1}{3}\)

\(\frac{1}{{\sqrt 3 }}\)

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

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

Questions 28 of 50

Question:The length of the latus rectum of an ellipse is \(\frac{1}{3}\) of the major axis. Its eccentricity is

Answers Choices:

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

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

\(\frac{{5 \times 4 \times 3}}{{{7^3}}}\)

\({\left( {\frac{3}{4}} \right)^4}\)

Questions 29 of 50

Question:The eccentricity of the ellipse \(4{x^2} + 9{y^2} = 36\), is

Answers Choices:

\(\frac{1}{{2\sqrt 3 }}\)

\(\frac{1}{{\sqrt 3 }}\)

\(\frac{{\sqrt 5 }}{3}\)

\(\frac{{\sqrt 5 }}{6}\)

Questions 30 of 50

Question:The eccentricity of the ellipse \(25{x^2} + 16{y^2} = 400\) is

Answers Choices:

3/5

1/3

2/5

1/5

Questions 31 of 50

Question:The equation \(14{x^2} - 4xy + 11{y^2} - 44x - 58y + 71 = 0\) represents

Answers Choices:

A circle

An ellipse

A hyperbola

A rectangular hyperbola

Questions 32 of 50

Question:The centre of the ellipse\(\frac{{{{(x + y - 2)}^2}}}{9} + \frac{{{{(x - y)}^2}}}{{16}} = 1\) is

Answers Choices:

(0, 0)

(1, 1)

(1, 0)

(0, 1)

Questions 33 of 50

Question:The position of the point (4, -3) with respect to the ellipse \(2{x^2} + 5{y^2} = 20\) is

Answers Choices:

Outside the ellipse

On the ellipse

On the major axis

None of these

Questions 34 of 50

Question:The equation of the tangent to the ellipse \({x^2} + 16{y^2} = 16\) making an angle of \({60^o}\)with x-axis is

Answers Choices:

\(\sqrt 3 x - y + 7 = 0\)

\(\sqrt 3 x - y - 7 = 0\)

\(\sqrt 3 x - y \pm 7 = 0\)

None of these

Questions 35 of 50

Question:The equation of the normal to the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) at the point \((a\cos \theta ,\;b\sin \theta )\) is

Answers Choices:

\(\frac{{ax}}{{\sin \theta }} - \frac{{by}}{{\cos \theta }} = {a^2} - {b^2}\)

\(\frac{{ax}}{{\sin \theta }} - \frac{{by}}{{\cos \theta }} = {a^2} + {b^2}\)

\(\frac{{ax}}{{\cos \theta }} - \frac{{by}}{{\sin \theta }} = {a^2} - {b^2}\)

\(\frac{{ax}}{{\cos \theta }} - \frac{{by}}{{\sin \theta }} = {a^2} + {b^2}\)

Questions 36 of 50

Question:If the normal at the point \(P(\theta )\) to the ellipse \(\frac{{{x^2}}}{{14}} + \frac{{{y^2}}}{5} = 1\) intersects it again at the point \(Q(2\theta )\), then \(\cos \theta \) is equal to

Answers Choices:

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

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

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

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

Questions 37 of 50

Question:The point (4, -3) with respect to the ellipse \(4{x^2} + 5{y^2} = 1\)

Answers Choices:

Lies on the curve

Is inside the curve

Is outside the curve

Is focus of the curve

Questions 38 of 50

Question:A point ratio of whose distance from a fixed point and line \(x = 9/2\) is always 2 : 3. Then locus of the point will be

Answers Choices:

Hyperbola

Ellipse

Parabola

Circle

Questions 39 of 50

Question:The locus of the point of intersection of the lines \(ax\sec \theta + by\tan \theta = a\) and \(ax\tan \theta + by\sec \theta = b\), where \(\theta \) is the parameter, is

Answers Choices:

A straight line

A circle

An ellipse

A hyperbola

Questions 40 of 50

Question:If the centre, vertex and focus of a hyperbola be (0, 0), (4, 0) and (6, 0) respectively, then the equation of the hyperbola is

Answers Choices:

\(4{x^2} - 5{y^2} = 8\)

\(4{x^2} - 5{y^2} = 80\)

\(5{x^2} - 4{y^2} = 80\)

\(5{x^2} - 4{y^2} = 8\)

Questions 41 of 50

Question:The eccentricity of the hyperbola \(2{x^2} - {y^2} = 6\) is

Answers Choices:

\(\sqrt 2 \)

2

3

\(\sqrt 3 \)

Questions 42 of 50

Question:The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is 6. The equation of the hyperbola referred to its axes as axes of co-ordinates is

Answers Choices:

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

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

\(3{x^2} - {y^2} = 9\)

\({x^2} - 3{y^2} = 9\)

Questions 43 of 50

Question:The eccentricity of the hyperbola \(5{x^2} - 4{y^2} + 20x + 8y = 4\) is

Answers Choices:

\(\sqrt 2 \)

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

2

3

Questions 44 of 50

Question:The latus rectum of the hyperbola \(9{x^2} - 16{y^2} + 72x - 32y - 16 = 0\) is

Answers Choices:

\(\frac{9}{2}\)

\( - \frac{9}{2}\)

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

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

Questions 45 of 50

Question:If the straight line \(x\cos \alpha + y\sin \alpha = p\) be a tangent to the hyperbola \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\), then

Answers Choices:

\({a^2}{\cos ^2}\alpha + {b^2}{\sin ^2}\alpha = {p^2}\)

\({a^2}{\cos ^2}\alpha - {b^2}{\sin ^2}\alpha = {p^2}\)

\({a^2}{\sin ^2}\alpha + {b^2}{\cos ^2}\alpha = {p^2}\)

\({a^2}{\sin ^2}\alpha - {b^2}{\cos ^2}\alpha = {p^2}\)

Questions 46 of 50

Question:If the tangent on the point \((2\sec \varphi ,\;3\tan \varphi )\) of the hyperbola \(\frac{{{x^2}}}{4} - \frac{{{y^2}}}{9} = 1\) is parallel to \(3x - y + 4 = 0\), then the value of \(\phi \) is

Answers Choices:

\({45^o}\)

\({60^o}\)

\({30^o}\)

\({75^o}\)

Questions 47 of 50

Question:The equation of the normal to the hyperbola \(\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1\) at \(( - 4,\;0)\) is

Answers Choices:

\(y = 0\)

\(y = x\)

\(x = 0\)

\(x = - y\)

Questions 48 of 50

Question:The eccentricity of the conjugate hyperbola of the hyperbola \({x^2} - 3{y^2} = 1\), is

Answers Choices:

2

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

4

\(\frac{4}{3}\)

Questions 49 of 50

Question:If \(5{x^2} + \lambda {y^2} = 20\) represents a rectangular hyperbola, then \(\lambda \) equals

Answers Choices:

5

4

-5

None of these

Questions 50 of 50

Question:The equation of the hyperbola referred to the axis as axes of co-ordinate and whose distance between the foci is 16 and eccentricity is \(\sqrt 2 \), is

Answers Choices:

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

\({x^2} - {y^2} = 32\)

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

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