## MCQ (Single Correct Answer)

Let $$\mathrm{P}\left(x_{0}, y_{0}\right)$$ be the point on the hyperbola $$3 x^{2}-4 y^{2}=36$$, which is nearest to the line $$3 x+2 y=1$$. Then $$\...

Let $\mathrm{H}$ be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :

If the maximum distance of normal to the ellipse $$\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b ...

Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2}, 0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then
$$S=\left\{x \in \mat...

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$. If one of these tangents touches the ...

The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$...

If $$\mathrm{P}(\mathrm{h}, \mathrm{k})$$ be a point on the parabola $$x=4 y^{2}$$, which is nearest to the point $$\mathrm{Q}(0,33)$$, then the dista...

If the tangent at a point P on the parabola $$y^2=3x$$ is parallel to the line $$x+2y=1$$ and the tangents at the points Q and R on the ellipse $$\fra...

The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of it...

Let T and C respectively be the transverse and conjugate axes of the hyperbola $$16{x^2} - {y^2} + 64x + 4y + 44 = 0$$. Then the area of the region ab...

The distance of the point $$(6,-2\sqrt2)$$ from the common tangent $$\mathrm{y=mx+c,m > 0}$$, of the curves $$x=2y^2$$ and $$x=1+y^2$$ is :

The equations of the sides AB and AC of a triangle ABC are $$(\lambda+1)x+\lambda y=4$$ and $$\lambda x+(1-\lambda)y+\lambda=0$$ respectively. Its ver...

Let a tangent to the curve $$\mathrm{y^2=24x}$$ meet the curve $$xy = 2$$ at the points A and B. Then the mid points of such line segments AB lie on a...

Let a line L pass through the point of intersection of the lines $$b x+10 y-8=0$$ and $$2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right...

Let the focal chord of the parabola $$\mathrm{P}: y^{2}=4 x$$ along the line $$\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$$ meet the parabola...

Let the hyperbola $$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ pass through the point $$(2 \sqrt{2},-2 \sqrt{2})$$. A parabola is drawn whose focu...

If the tangents drawn at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the parabola $$y^{2}=2 x-3$$ intersect at the point $$R(0,1)$$, then the orth...

If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x+y=a$$, is 16, then $$|a|$$ is equal to...

Let $$P(a, b)$$ be a point on the parabola $$y^{2}=8 x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^{2}+y^{2}-10 x-14 ...

Let $$\mathrm{P}$$ and $$\mathrm{Q}$$ be any points on the curves $$(x-1)^{2}+(y+1)^{2}=1$$ and $$y=x^{2}$$, respectively. The distance between $$P$$ ...

The acute angle between the pair of tangents drawn to the ellipse $$2 x^{2}+3 y^{2}=5$$ from the point $$(1,3)$$ is

The equation of a common tangent to the parabolas $$y=x^{2}$$ and $$y=-(x-2)^{2}$$ is

If the line $$x-1=0$$ is a directrix of the hyperbola $$k x^{2}-y^{2}=6$$, then the hyperbola passes through the point

Let the tangent drawn to the parabola $$y^{2}=24 x$$ at the point $$(\alpha, \beta)$$ is perpendicular to the line $$2 x+2 y=5$$. Then the normal to t...

If the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ meets the line $$\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$$ on the $$x$$-axis and the line $$\f...

The tangents at the points $$A(1,3)$$ and $$B(1,-1)$$ on the parabola $$y^{2}-2 x-2 y=1$$ meet at the point $$P$$. Then the area (in unit $${ }^{2}$$ ...

Let the foci of the ellipse $$\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$$ and the hyperbola $$\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$$ coincide....

Let the eccentricity of the ellipse $${x^2} + {a^2}{y^2} = 25{a^2}$$ be b times the eccentricity of the hyperbola $${x^2} - {a^2}{y^2} = 5$$, where a ...

Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of $${\pi \over 4}$$ with the line y = 3x + 5 to...

Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of $${\pi \over 2}$$ at the point (3, 0). Let the line segment PQ be a...

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \o...

If vertex of a parabola is (2, $$-$$1) and the equation of its directrix is 4x $$-$$ 3y = 21, then the length of its latus rectum is :

Let the eccentricity of the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $$\sqrt {{5 \over 2}} $$ and length of its latus ...

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $$3x + y - 29 = 0$$, is $${x^2} + a{y^2} + bxy + cx + dy + k = 0$$, th...

Let the eccentricity of an ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $$a > b$$, be $${1 \over 4}$$. If this ellipse passes th...

If m is the slope of a common tangent to the curves $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$ and $${x^2} + {y^2} = 12$$, then $$12{m^2}$$ is e...

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse $${x^2} + 2{y^2} = 4$$ is an ellipse with eccent...

The normal to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1$$ at the point $$\left( {8,3\sqrt 3 } \right)$$ on it passes through the...

Let the normal at the point on the parabola y2 = 6x pass through the point (5, $$-$$8). If the tangent at P to the parabola intersects its directrix a...

If the line $$y = 4 + kx,\,k > 0$$, is the tangent to the parabola $$y = x - {x^2}$$ at the point P and V is the vertex of the parabola, then the slop...

The line y = x + 1 meets the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$$ at two points P and Q. If r is the radius of the circle with PQ as ...

If $$y = {m_1}x + {c_1}$$ and $$y = {m_2}x + {c_2}$$, $${m_1} \ne {m_2}$$ are two common tangents of circle $${x^2} + {y^2} = 2$$ and parabola y2 = x,...

Let $$x = 2t$$, $$y = {{{t^2}} \over 3}$$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $$SA \bot BA$$, where A...

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q...

Let the maximum area of the triangle that can be inscribed in the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$$, having one of i...

Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to ___________....

Consider the parabola with vertex $$\left( {{1 \over 2},{3 \over 4}} \right)$$ and the directrix $$y = {1 \over 2}$$. Let P be the point where the par...

Let $$\theta$$ be the acute angle between the tangents to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 1} = 1$$ and the circle $${x^2} + {y^2} = 3...

The locus of mid-points of the line segments joining ($$-$$3, $$-$$5) and the points on the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$$ is :

An angle of intersection of the curves, $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ and x2 + y2 = ab, a > b, is :...

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the o...

The line $$12x\cos \theta + 5y\sin \theta = 60$$ is tangent to which of the following curves?

If two tangents drawn from a point P to the parabola y2 = 16(x $$-$$ 3) are at right angles, then the locus of point P is :

A tangent and a normal are drawn at the point P(2, $$-$$4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B res...

If x2 + 9y2 $$-$$ 4x + 3 = 0, x, y $$\in$$ R, then x and y respectively lie in the intervals :

The point $$P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)$$ lies on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ having eccentric...

The locus of the mid points of the chords of the hyperbola x2 $$-$$ y2 = 4, which touch the parabola y2 = 8x, is :

On the ellipse $${{{x^2}} \over 8} + {{{y^2}} \over 4} = 1$$ let P be a point in the second quadrant such that the tangent at P to the ellipse is perp...

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix...

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line ...

If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes...

The locus of the centroid of the triangle formed by any point P on the hyperbola $$16{x^2} - 9{y^2} + 32x + 36y - 164 = 0$$, and its foci is :

Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents a...

Let an ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $${a^2} > {b^2}$$, passes through $$\left( {\sqrt {{3 \over 2}} ,1} \ri...

Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $$-$$ y2 = 3. If L is also a tangent to the parabola y2 = $$\alpha$$x, then $$\al...

Let $${E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b$$. Let E2 be another ellipse such that it touches the end points of major a...

Let P be a variable point on the parabola $$y = 4{x^2} + 1$$. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn ...

Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the ar...

Consider a hyperbola H : x2 $$-$$ 2y2 = 4. Let the tangent at a point P(4, $${\sqrt 6 }$$) meet the x-axis at Q and latus rectum at R(x1, y1), x1 >...

Let a tangent be drawn to the ellipse $${{{x^2}} \over {27}} + {y^2} = 1$$ at $$(3\sqrt 3 \cos \theta ,\sin \theta )$$ where $$0 \in \left( {0,{\pi \...

Let L be a tangent line to the parabola y2 = 4x $$-$$ 20 at (6, 2). If L is also a tangent to the ellipse $${{{x^2}} \over 2} + {{{y^2}} \over b} = 1$...

If the points of intersections of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$ and the circle x2 + y2 = 4b, b > 4 lie on the ...

Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1)...

If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a $$\ne$$ 0, then 'a' must be greater than :

The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, $${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$...

The shortest distance between the line x $$-$$ y = 1 and the curve x2 = 2y is :

A hyperbola passes through the foci of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$ and its transverse and conjugate axes coincide ...

A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?

If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x $$-$$ 1, then the co-ordinates of P are :

The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a
moving point of the parabola, is another parabola whose...

If the normal at an end of a latus rectum of an
ellipse passes through an extremity of the
minor axis, then the eccentricity e of the ellipse
satisfie...

The centre of the circle passing through the
point (0, 1) and touching the parabola y = x2 at the point (2, 4) is :

Let L1
be a tangent to the parabola y2 = 4(x + 1) and L2
be a tangent to the parabola
y2 = 8(x + 2) such that L1
and L2
intersect at right angles....

Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent
to the ellipse,
$${{{x^2}} \over 4} + {{{y^2}} \ove...

If the line y = mx + c is a common tangent to
the hyperbola
$${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$$ and the circle
x2
+ y2
= 36, then w...

If the common tangent to the parabolas, y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal to :...

If the co-ordinates of two points A and B are $$\left( {\sqrt 7 ,0} \right)$$ and $$\left( { - \sqrt 7 ,0} \right)$$ respectively and P is any
point o...

Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is $${1 \over 2}$$. If P(1, $$\beta $$), $$\beta $$ > 0 i...

Let P(3, 3) be a point on the hyperbola, $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal to it at P intersects the x-axis
at ...

Let $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity i...

Let the latus ractum of the parabola y2
= 4x be
the common chord to the circles C1
and C2
each of them having radius 2$$\sqrt 5 $$. Then, the
distan...

Let e1
and e2
be the eccentricities of the
ellipse, $${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$$(b < 5) and the hyperbola,
$${{{x^2}} \...

A hyperbola having the transverse axis of
length
$$\sqrt 2 $$ has the same foci as that of the ellipse
3x2 + 4y2 = 12, then this hyperbola does not
pa...

Let P be a point on the parabola, y2
= 12x and
N be the foot of the perpendicular drawn from
P on the axis of the parabola. A line is now
drawn throu...

The area (in sq. units) of an equilateral triangle
inscribed in the parabola y2 = 8x, with one of
its vertices on the vertex of this parabola, is :...

For some $$\theta \in \left( {0,{\pi \over 2}} \right)$$, if the eccentricity of the
hyperbola, x2–y2sec2$$\theta $$ = 10 is
$$\sqrt 5 $$ times the
...

A line parallel to the straight line 2x – y = 0 is
tangent to the hyperbola
$${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$$ at the point
$$\left( {{x_1...

The length of the minor axis (along y-axis) of
an ellipse in the standard form is $${4 \over {\sqrt 3 }}$$. If this
ellipse touches the line, x + 6y =...

If one end of a focal chord AB of the parabola
y2 = 8x is at $$A\left( {{1 \over 2}, - 2} \right)$$, then the equation of
the tangent to it at B is :...

If e1 and e2 are the eccentricities of the ellipse,
$${{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1$$ and the hyperbola, $${{{x^2}} \over 9} - {{{y^2}}...

The length of the perpendicular from the origin,
on the normal to the curve, x2 + 2xy – 3y2 = 0
at the point (2,2) is

If a hyperbola passes through the point
P(10, 16) and it has vertices at (± 6, 0), then the
equation of the normal to it at P is

Let the line y = mx and the ellipse 2x2 + y2 = 1
intersect at a ponit P in the first quadrant. If the
normal to this ellipse at P meets the co-ordinat...

The locus of a point which divides the line
segment joining the point (0, –1) and a point on
the parabola, x2 = 4y, internally in the ratio
1 : 2, is ...

If 3x + 4y = 12$$\sqrt 2 $$ is a tangent to the ellipse
$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1$$ for some $$a$$ $$ \in $$ R, then the dista...

If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to:

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12,
then the length of its latus rectum is:

The equation of common tangent to the curves y2
= 16x and xy = –4, is :

The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3, intersect at the point :

An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?

Let P be the point of intersection of the common tangents to the parabola y2
= 12x and the hyperbola
8x2
– y2
= 8. If S and S' denote the foci of th...

If the normal to the ellipse 3x2
+ 4y2
= 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent
to the ellipse at P passes throug...

If 5x + 9 = 0 is the directrix of the hyperbola 16x2
– 9y2
= 144, then its corresponding focus is :

If the line ax + y = c, touches both the curves x2
+ y2
= 1 and y2
= 4$$\sqrt 2 $$x , then |c| is equal to :

The tangent and normal to the ellipse 3x2
+ 5y2
= 32 at the point P(2, 2) meet the x-axis at Q and R,
respectively. Then the area (in sq. units) of ...

If a directrix of a hyperbola centred at the origin and passing through the point (4, –2$$\sqrt 3 $$ ) is 5x = 4$$\sqrt 5 $$ and
its eccentricity is e...

If the line x – 2y = 12 is tangent to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ at the point $$\left( {3, - {9 \over 2}} \...

If the tangent to the parabola y2 = x at a point
($$\alpha $$, $$\beta $$), ($$\beta $$ > 0) is also a tangent to the ellipse,
x2 + 2y2 = 1, then $...

The area (in sq. units) of the smaller of the two
circles that touch the parabola, y2 = 4x at the point
(1, 2) and the x-axis is :-

If the line y = mx + 7$$\sqrt 3 $$ is normal to the
hyperbola
$${{{x^2}} \over {24}} - {{{y^2}} \over {18}} = 1$$ , then a value of m is

If one end of a focal chord of the parabola,
y2 = 16x is at (1, 4), then the length of this focal
chord is

If the eccentricity of the standard hyperbola
passing through the point (4,6) is 2, then the
equation of the tangent to the hyperbola at (4,6)
is

The tangent to the parabola y2
= 4x at the point
where it intersects the circle x2
+ y2
= 5 in the
first quadrant, passes through the point :
...

In an ellipse, with centre at the origin, if the
difference of the lengths of major axis and minor
axis is 10 and one of the foci is at (0,5$$\sqrt 3$...

The shortest distance between the line y = x and
the curve y2 = x – 2 is :

If the tangents on the ellipse 4x2 + y2 = 8 at the
points (1, 2) and (a, b) are perpendicular to each
other, then a2 is equal to :
...

The equation of a tangent to the parabola, x2
= 8y, which makes an angle $$\theta $$ with the positive directions of x-axis, is :

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $$\Delta $$S'BS is a right angled triangle with right...

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the re...

If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the following points does not lie on this...

Let P(4, –4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this ...

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the
eccentricity of the hyperbola is :

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of ...

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of...

If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the othertwo vertices are the points of intersec...

Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is

If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents inte...

The length of the chord of the parabola x2 $$=$$ 4y having equation x – $$\sqrt 2 y + 4\sqrt 2 = 0$$ is -

Let S = $$\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1.$$ Then S represents

If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c) ...

The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is -

Let A(4, $$-$$ 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the...

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x-axis. Then the eccentricity ...

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the
origin, on the positive x-axis then which...

Let $$0 < \theta < {\pi \over 2}$$. If the eccentricity of the
hyperbola $${{{x^2}} \over {{{\cos }^2}\theta }} - {{{y^2}} \over {{{\sin }^2}...

Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x is :

If $$\theta $$ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, the |tan $$\theta $$| is equa...

The locus of the point of intersection of the lines, $$\sqrt 2 x - y + 4\sqrt 2 k = 0$$ and $$\sqrt 2 k\,x + k\,y - 4\sqrt 2 = 0$$ (k is any non-zero...

Let P be a point on the parabola, x2 = 4y. If the distance of P from the center of the circle, x2 + y2 + 6x + 8 = 0 is minimum, then the equation of ...

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus an its nearest vertex on the major axis is $${3 \over 2}$$...

Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q.
If these tangents intersect at the
point T(0, 3) then the area (in sq. units)...

Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the
parabola at A and B, respectively. If C is the cen...

Tangents drawn from the point ($$-$$8, 0) to the parabola y2 = 8x touch the parabola at $$P$$ and $$Q.$$ If F is the focus of the parabola, then the a...

A normal to the hyperbola, 4x2 $$-$$ 9y2 = 36 meets the co-ordinate axes $$x$$ and y at A and B, respectively. If the parallelogram OABP (O being the ...

If the tangents drawn to the hyperbola 4y2 = x2 + 1 intersect the co-ordinate axes at the distinct points A and B then the locus of the mid point of A...

Two parabolas with a common vertex and with axes along x-axis and $$y$$-axis, respectively intersect each other in the first quadrant. If the length o...

If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then $$\left| c \right|$$ is equal to :

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate
axes and passing through the points (4, −1) and (−2, 2) is :

Let z$$ \in $$C, the set of complex numbers. Then the equation, 2|z + 3i| $$-$$ |z $$-$$ i| = 0 represents :

The locus of the point of intersection of the straight lines,
tx $$-$$ 2y $$-$$ 3t = 0
x $$-$$ 2ty + 3 = 0 (t $$ \in $$ R), is : ...

If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles $${\cos ^{ - 1}}\left( {{1 \ove...

If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance of P from the origin, is : ...

Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is $${3 \over 5}$$ and the distance bet...

A hyperbola passes through the point P$$\left( {\sqrt 2 ,\sqrt 3 } \right)$$ and has foci at $$\left( { \pm 2,0} \right)$$. Then the tangent to this h...

The eccentricity of an ellipse whose centre is at the origin is $${1 \over 2}$$. If one of its directrices is x = – 4, then the
equation of the normal...

P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P passes through Q, then the minimum...

A hyperbola whose transverse axis is along the major axis of the conic, $${{{x^2}} \over 3} + {{{y^2}} \over 4} = 4$$ and has vertices at the foci of ...

If the tangent at a point on the ellipse $${{{x^2}} \over {27}} + {{{y^2}} \over 3} = 1$$ meets the coordinate axes at A and B, and O is the origin, t...

Let a and b respectively be the semitransverse and semi-conjugate axes of a
hyperbola whose eccentricity satisfies the equation 9e2 − 18e + 5 = 0. If ...

The eccentricity of the hyperbola whose length of the latus rectum is equal to $$8$$ and the length of its conjugate axis is equal to half of the dist...

Let $$P$$ be the point on the parabola, $${{y^2} = 8x}$$ which is at a minimum distance from the centre $$C$$ of the circle, $${x^2} + {\left( {y + 6}...

The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0$$, at $$(1,1)$$

Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $${{x^2} = 8y}$$. If the point $$P$$ divides the line segment $$OQ$$ internally in the...

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $${{{x^2}} \over 9} + {{{y^2}...

The locus of the foot of perpendicular drawn from the centre of the ellipse $${x^2} + 3{y^2} = 6$$ on any tangent to it is

The slope of the line touching both the parabolas $${y^2} = 4x$$ and $${x^2} = - 32y$$ is

The equation of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and having centre at $$(0,3)$$ is...

Given : A circle, $$2{x^2} + 2{y^2} = 5$$ and a parabola, $${y^2} = 4\sqrt 5 x$$.
Statement-1 : An equation of a common tangent to these curves is $$...

STATEMENT-1 : An equation of a common tangent to the parabola $${y^2} = 16\sqrt 3 x$$ and the ellipse $$2{x^2} + {y^2} = 4$$ is $$y = 2x + 2\sqrt 3 $$...

An ellipse is drawn by taking a diameter of thec circle $${\left( {x - 1} \right)^2} + {y^2} = 1$$ as its semi-minor axis and a diameter of the circle...

Equation of the ellipse whose axes of coordinates and which passes through the point $$(-3,1)$$ and has eccentricity $$\sqrt {{2 \over 5}} $$ is

If two tangents drawn from a point $$P$$ to the parabola $${y^2} = 4x$$ are at right angles, then the locus of $$P$$ is

The ellipse $${x^2} + 4{y^2} = 4$$ is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscribed in another ellipse that pa...

A focus of an ellipse is at the origin. The directrix is the line $$x=4$$ and the eccentricity is $${{1 \over 2}}$$. Then the length of the semi-major...

A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at

For the Hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$ , which of the following remains constant when $...

The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y=x+2$$. The point on this line from which the other tangent to the parabola is perpendi...

The normal to a curve at $$P(x,y)$$ meets the $$x$$-axis at $$G$$. If the distance of $$G$$ from the origin is twice the abscissa of $$P$$, then the c...

In the ellipse, the distance between its foci is $$6$$ and minor axis is $$8$$. Then its eccentricity is

The locus of the vertices of the family of parabolas
$$y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a$$ is

Angle between the tangents to the curve $$y = {x^2} - 5x + 6$$ at the points $$(2,0)$$ and $$(3,0)$$ is

An ellipse has $$OB$$ as semi minor axis, $$F$$ and $$F$$' its focii and theangle $$FBF$$' is a right angle. Then the eccentricity of the ellipse is

The locus of a point $$P\left( {\alpha ,\beta } \right)$$ moving under the condition that the line $$y = \alpha x + \beta $$ is tangent to the hyperbo...

Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the parabola $${y^2} = 8x$$. The locus of mid point of $$PQ$$ is

If $$a \ne 0$$ and the line $$2bx+3cy+4d=0$$ passes through the points of intersection of the parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$, then

The eccentricity of an ellipse, with its centre at the origin, is $${1 \over 2}$$. If one of the directrices is $$x=4$$, then the equation of the elli...

The normal at the point$$\left( {bt_1^2,2b{t_1}} \right)$$ on a parabola meets the parabola again in the point $$\left( {bt_2^2,2b{t_2}} \right)$$, th...

The foci of yhe ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$ and the hyperbola $${{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \...

Two common tangents to the circle $${x^2} + {y^2} = 2{a^2}$$ and parabola $${y^2} = 8ax$$ are

## Numerical

If the $$x$$-intercept of a focal chord of the parabola $$y^{2}=8x+4y+4$$ is 3, then the length of this chord is equal to ____________.

The line $$x=8$$ is the directrix of the ellipse $$\mathrm{E}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ with the corresponding focus $$(2,0)$$. If t...

A triangle is formed by the tangents at the point (2, 2) on the curves $$y^2=2x$$ and $$x^2+y^2=4x$$, and the line $$x+y+2=0$$. If $$r$$ is the radius...

The vertices of a hyperbola H are ($$\pm$$ 6, 0) and its eccentricity is $${{\sqrt 5 } \over 2}$$. Let N be the normal to H at a point in the first qu...

Let C be the largest circle centred at (2, 0) and inscribed in the ellipse $${{{x^2}} \over {36}} + {{{y^2}} \over {16}} = 1$$. If (1, $$\alpha$$) lie...

Let a tangent to the curve $$9{x^2} + 16{y^2} = 144$$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segmen...

Let the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the ellipse $$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$$ meet at the point $$R(\sqrt{2}, ...

Two tangent lines $$l_{1}$$ and $$l_{2}$$ are drawn from the point $$(2,0)$$ to the parabola $$2 \mathrm{y}^{2}=-x$$. If the lines $$l_{1}$$ and $$l_{...

For the hyperbola $$\mathrm{H}: x^{2}-y^{2}=1$$ and the ellipse $$\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$$, a $$>\ma...

A common tangent $$\mathrm{T}$$ to the curves $$\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$ and $$C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$...

An ellipse $$E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ passes through the vertices of the hyperbola $$H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$$....

If the length of the latus rectum of the ellipse $$x^{2}+4 y^{2}+2 x+8 y-\lambda=0$$ is 4 , and $$l$$ is the length of its major axis, then $$\lambda+...

Let the equation of two diameters of a circle $$x^{2}+y^{2}-2 x+2 f y+1=0$$ be $$2 p x-y=1$$ and $$2 x+p y=4 p$$. Then the slope m $$ \in $$ $$(0, \in...

The sum of diameters of the circles that touch (i) the parabola $$75 x^{2}=64(5 y-3)$$ at the point $$\left(\frac{8}{5}, \frac{6}{5}\right)$$ and (ii)...

Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $$\...

Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the c...

A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola $$y = {\left( {x - {1 \over 4}} \ri...

Let a line L1 be tangent to the hyperbola $${{{x^2}} \over {16}} - {{{y^2}} \over 4} = 1$$ and let L2 be the line passing through the origin and perpe...

Let the common tangents to the curves $$4({x^2} + {y^2}) = 9$$ and $${y^2} = 4x$$ intersect at the point Q. Let an ellipse, centered at the origin O, ...

Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $${5 \over 4}$$. If the equation of the normal at t...

Let the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {y^2} = 1$$ and the ellipse $$E:3{x^2} + 4{y^2} = 12$$ be such that the length of latus rectum of H is...

Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x...

If two tangents drawn from a point ($$\alpha$$, $$\beta$$) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one ...

A tangent line L is drawn at the point (2, $$-$$4) on the parabola y2 = 8x. If the line L is also tangent to the circle x2 + y2 = a, then 'a' is equa...

Let A (sec$$\theta$$, 2tan$$\theta$$) and B (sec$$\phi$$, 2tan$$\phi$$), where $$\theta$$ + $$\phi$$ = $$\pi$$/2, be two points on the hyperbola 2x2 $...

If the minimum area of the triangle formed by a tangent to the ellipse $${{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2}}} = 1$$ and the co-ordinate ...

Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, $$-$$4), one focus at (4, $$-$$4) and one vertex at (5,...

If the point on the curve y2 = 6x, nearest to the point $$\left( {3,{3 \over 2}} \right)$$ is ($$\alpha$$, $$\beta$$), then 2($$\alpha$$ + $$\beta$$) ...

Let y = mx + c, m > 0 be the focal chord of y2 = $$-$$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4$$\sqrt 2 $$ (m + c) is equ...

Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the square of the slope of the line L is __________....

A line is a common tangent to the circle (x $$-$$ 3)2 + y2 = 9 and the parabola y2 = 4x. If the two points of contact (a, b) and (c, d) are distinct a...

The locus of the point of intersection of the lines $$\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0$$ and $$\sqrt 3 x - y - 4\left( {\sqrt 3 } \rig...

If the tangent to the curve, y = ex
at a point
(c, ec) and the normal to the parabola, y2 = 4x
at the point (1, 2) intersect at the same point on
the...

Let a line y = mx (m > 0) intersect the parabola,
y2 = x at a point P, other than the origin. Let
the tangent to it at P meet the x-axis at the poi...