Coordinate Geometry - Conic Section Test 3

Total Questions:50 Total Time: 75 Min

Remaining:

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

$$9{y^2} = 4ax$$

$$9{x^2} = 4ay$$

$$9{y^2} + 4ax = 0$$

$$9{x^2} + 4ay = 0$$

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

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

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

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

$${y^2} = 12x$$

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

Questions 4 of 50

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

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

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

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

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

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

Questions 6 of 50

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

(0, 2)

(1, 2)

(2, 0)

(2, 1)

Questions 7 of 50

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

(–2, 11/2)

(–2, 2)

(–2, 11)

(2, 11)

Questions 8 of 50

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

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

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

(1, 0)

(0, 1)

Questions 9 of 50

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

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

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

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

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

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

Questions 11 of 50

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

$${x^2} + 4x - 8y - 12 = 0$$

$${x^2} - 4x + 8y + 12 = 0$$

$${x^2} + 8y = 12$$

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

Questions 12 of 50

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

$$x = 1$$

$$y = 0$$

$$x = - 1$$

$$y = - 1$$

Questions 13 of 50

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

$$\lambda < \frac{1}{4}$$

$$\lambda > \frac{1}{4}$$

$$\lambda = 4$$

$$\lambda = 1$$

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

$$yt = x + a{t^2}$$

$$y = xt + a{t^2}$$

$$y = xt + \frac{a}{t}$$

$$y = tx$$

Questions 15 of 50

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

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

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

$$\frac{1}{3}$$

4

Questions 16 of 50

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

$$a = c$$

$$\frac{a}{c} = m$$

$$m = {a^2}c$$

$$m = a{c^2}$$

Questions 17 of 50

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

$$\frac{\pi }{6}$$

$$\frac{\pi }{4}$$

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

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

$$\frac{1}{2}\sqrt {41}$$

$$\sqrt {41}$$

$$\frac{3}{2}\sqrt {41}$$

$$2\sqrt {41}$$

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

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

$$\{ - 1,\;1\}$$

{–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

{–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

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

$${y^2} = x$$

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

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

(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

$$9{x^2} + 25{y^2} = 225$$

$$25{x^2} + 9{y^2} = 225$$

$$3{x^2} + 4{y^2} = 192$$

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

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

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

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

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

$$\frac{1}{3}$$

$$\frac{1}{{\sqrt 3 }}$$

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

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

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

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

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

$$\frac{{5 \times 4 \times 3}}{{{7^3}}}$$

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

Questions 29 of 50

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

$$\frac{1}{{2\sqrt 3 }}$$

$$\frac{1}{{\sqrt 3 }}$$

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

$$\frac{{\sqrt 5 }}{6}$$

Questions 30 of 50

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

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

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

(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

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

$$\sqrt 3 x - y + 7 = 0$$

$$\sqrt 3 x - y - 7 = 0$$

$$\sqrt 3 x - y \pm 7 = 0$$

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

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

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

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

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

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

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

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

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

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

Questions 37 of 50

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

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

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

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

$$4{x^2} - 5{y^2} = 8$$

$$4{x^2} - 5{y^2} = 80$$

$$5{x^2} - 4{y^2} = 80$$

$$5{x^2} - 4{y^2} = 8$$

Questions 41 of 50

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

$$\sqrt 2$$

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

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

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

$$3{x^2} - {y^2} = 9$$

$${x^2} - 3{y^2} = 9$$

Questions 43 of 50

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

$$\sqrt 2$$

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

2

3

Questions 44 of 50

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

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

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

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

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

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

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

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

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

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

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

$${45^o}$$

$${60^o}$$

$${30^o}$$

$${75^o}$$

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

$$y = 0$$

$$y = x$$

$$x = 0$$

$$x = - y$$

Questions 48 of 50

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

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

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

$${x^2} - {y^2} = 16$$
$${x^2} - {y^2} = 32$$
$${x^2} - 2{y^2} = 16$$
$${y^2} - {x^2} = 16$$