Mathematics
Conic Sections
Previous Years Questions

## Numerical

Consider the hyperbola $$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$$ with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be ...
Let E be the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$. For any three distinct points P, Q and Q' on E, let M(P, Q) be the mid-point of...
For how many values of p, the circle x2 + y2 + 2x + 4y $$-$$ p = 0 and the coordinate axes have exactly three common points?
Suppose that the foci of the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1$$ are $$\left( {{f_1},0} \right)$$ and $$\left( {{f_2},0} \right)$$ w...
If the normals of the parabola $${y^2} = 4x$$ drawn at the end points of its latus rectum are tangents to the circle $${\left( {x - 3} \right)^2} + {\... Let the curve$$C$$be the mirror image of the parabola$${y^2} = 4x$$with respect to the line$$x+y+4=0$$. If$$A$$and$$B$$are the points of inte... A vertical line passing through the point$$(h,0)$$intersects the ellipse$${{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$$at the points$$P$$and$$Q$$... Let$$S$$be the focus of the parabola$${y^2} = 8x$$and let$$PQ$$be the common chord of the circle$${x^2} + {y^2} - 2x - 4y = 0$$and the given p... Consider the parabola$${y^2} = 8x$$. Let$${\Delta _1}$$be the area of the triangle formed by the end points of its latus rectum and the point$$P\l...
The line $$2x + y = 1$$ is tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If this line passes through the point o...

## MCQ (More than One Correct Answer)

Consider the parabola $$y^{2}=4 x$$. Let $$S$$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $$P=(-2,1)$$ meet...
Let E denote the parabola y2 = 8x. Let P = ($$-$$2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E....
Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola $${{{x^2}} \ov... Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :$${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1...
Let T be the line passing through the points P($$-$$2, 7) and Q(2, $$-$$5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent t...
Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the po...
If $$2x - y + 1 = 0$$ is a tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {16}} = 1$$ then which of the following CANNOT be sides...
If a chord, which is not a tangent, of the parabola y2 = 16x has the equation 2x + y = p, and mid-point (h, k), then which of the following is(are) po...
Let $$P$$ be the point on the parabola $${y^2} = 4x$$ which is at the shortest distance from the center $$S$$ of the circle $${x^2} + {y^2} - 4x - 16y... The circle$${C_1}:{x^2} + {y^2} = 3,$$with centre at$$O$$, intersects the parabola$${x^2} = 2y$$at the point$$P$$in the first quadrant, Let the... Let$${E_1}$$and$${E_2}$$be two ellipses whose centres are at the origin. The major axes of$${E_1}$$and$${E_2}$$lie along the$$x$$-axis and th... Consider the hyperbola$$H:{x^2} - {y^2} = 1$$and a circle$$S$$with center$$N\left( {{x_2},0} \right)$$. Suppose that$$H$$and$$S$$touch each o... Let$$P$$and$$Q$$be distinct points on the parabola$${y^2} = 2x$$such that a circle with$$PQ$$as diameter passes through the vertex$$O$$of th... Tangents are drawn to the hyperbola$${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1,$$parallel to the straight line$$2x - y = 1,$$The points of contact... Let the eccentricity of the hyperbola$${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$be reciprocal to that of the ellipse$${x^2} + 4{y^2} ...
Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by
Let $$A$$ and $$B$$ be two distinct points on the parabola $${y^2} = 4x$$. If the axis of the parabola touches a circle of radius $$r$$ having $$AB$$ ...
The tangent $$PT$$ and the normal $$PN$$ to the parabola $${y^2} = 4ax$$ at a point $$P$$ on it meet its axis at points $$T$$ and $$N$$, respectively....
In a triangle $$ABC$$ with fixed base $$BC$$, the vertex $$A$$ moves such that $$\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.$$$If $$a, b$$ and $$c... An ellipse intersects the hyperbola$$2{x^2} - 2{y^2} = 1$$orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If t... Let$$P\left( {{x_1},{y_1}} \right)$$and$$Q\left( {{x_2},{y_2}} \right),{y_1} < 0,{y_2} < 0,$$be the end points of the latus rectum of the el... Let a hyperbola passes through the focus of the ellipse$${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$. The transverse and conjugate axes of this... The equations of the common tangents to the parabola$$y = {x^2}$$and$$y = - {\left( {x - 2} \right)^2}$$is/are On the ellipse$$4{x^2} + 9{y^2} = 1,$$the points at which the tangents are parallel to the line$$8x = 9y$$are ## MCQ (Single Correct Answer) Consider the ellipse$$$ \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 $$Let H(\alpha, 0), 0 .tg {border-collapse:collapse;border-spacing:0;} .tg td{border... Let a, b and$$\lambda $$be positive real numbers. Suppose P is an end point of the latus return of the parabola y2 = 4$$\lambda $$x, and suppose the... Let the circles C1 : x2 + y2 = 9 and C2 : (x$$-$$3)2 + (y$$-$$4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x$$-$$... Let the circle C1 : x2 + y2 = 9 and C2 : (x$$-$$3)2 + (y$$-$$4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x$$-$$... Let$$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtend... Let S be the circle in the XY-plane defined the equation x2 + y2 = 4.Let E1E2 and F1F2 be the chords of S passing through the point P0 (1, 1) and para... Let S be the circle in the XY-plane defined the equation x2 + y2 = 4.Let P be a point on the circle S with both coordinates being positive. Let the ta... For$$a = \sqrt 2 $$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact ($$-$$1, 1), then which of the following options is... The tangent to a suitable conic (Column 1) at$$\left( {\sqrt 3 ,\,{1 \over 2}} \right)$$is found to be$$\sqrt 3 x + 2y = 4$$, then which of the fol... If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the o... Let$${F_1}\left( {{x_1},0} \right)$$and$${F_2}\left( {{x_2},0} \right)$$for$${{x_1} < 0}$$and$${{x_2} > 0}$$, be the foci of the ellipse ... Let$${F_1}\left( {{x_1},0} \right)$$and$${F_2}\left( {{x_2},0} \right)$$for$${{x_1} < 0}$$and$${{x_2} > 0}$$, be the foci of the ellipse ... The common tangents to the circle$${x^2} + {y^2} = 2$$and the parabola$${y^2} = 8x$$touch the circle at the points$$P, Q$$and the parabola at th... Let$$a, r, s, t$$be nonzero real numbers. Let$$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$and$$S\,\,\left( {a...
Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$ and $$S\,\,\left( {a... A line$$L:y=mx+3$$meets$$y$$-axis at$$(E, 3)$$and the are of the parabola$${y^2} = 16x,0 \le y \le 6$$at the point$$F\left( {{x_0},{y_0}}...
Let $$PQ$$ be a focal chord of the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$P$$ and $$Q$$ meet at a point lying on the line $$y=2x+... Let$$PQ$$be a focal chord of the parabola$${y^2} = 4ax$$. The tangents to the parabola at$$P$$and$$Q$$meet at a point lying on the line$$y=2x+...
The ellipse $${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ is inscribed in a rectangle $$R$$ whose sides are parallel to the coordinate axes. Ano...
Let $$(x, y)$$ be any point on the parabola $${y^2} = 4x$$. Let $$P$$ be the point that divides the line segment from $$(0, 0)$$ to $$(x, y)$$ in the ...
Let $$P(6, 3)$$ be a point on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal at the point $$P$$ intersects the...
The circle $${x^2} + {y^2} - 8x = 0$$ and hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ intersect at the points $$A$$ and $$B$$. Equation of...
The circle $${x^2} + {y^2} - 8x = 0$$ and hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ intersect at the points $$A$$ and $$B$$. Equation of...
Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ellipse at points $$A$$ and $$... Tangents are drawn from the point$$P(3, 4)$$to the ellipse$${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$touching the ellipse at points$$A$$and$$...
Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ellipse at points $$A$$ and $$... The line passing through the extremity$$A$$of the major axis and extremity$$B$$of the minor axis of the ellipse$${x^2} + 9{y^2} = 9$$meets its a... The normal at a point$$P$$on the ellipse$${x^2} + 4{y^2} = 16$$meets the$$x$$- axis$$Q$$. If$$M$$is the mid point of the line segment$$PQ$$, ... The locus of the orthocentre of the triangle formed by the lines$$$\left( {1 + p} \right)x - py + p\left( {1 + p} \right) = 0,$\left( {1 + q} ...
Let $$a$$ and $$b$$ be non-zero real numbers. Then, the equation $${x^2}{\cos ^2}\theta - {y^2}{\sin ^2}\theta = 0$$ represents
Consider a branch of the hyperbola $${x^2} - 2{y^2} - 2\sqrt 2 x - 4\sqrt 2 y - 6 = 0$$$with vertex at the point $$A$$. Let $$B$$ be one of the en... A hyperbola, having the transverse axis of length $$2\sin \theta ,$$ is confocal with the ellipse $$3{x^2} + 4{y^2} = 12.$$ Then its equation is Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at $$P$$ and $$Q$$ in the first and the fourth quadrants, re... Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at $$P$$ and $$Q$$ in the first and the fourth quadrants, re... Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at $$P$$ and $$Q$$ in the first and the fourth quadrants, re... STATEMENT-1: The curve $$y = {{ - {x^2}} \over 2} + x + 1$$ is symmetric with respect to the line $$x=1$$. because STATEMENT-2: A parabola is symmetri... The axis of a parabola is along the line $$y = x$$ and the distances of its vertex and focus from origin are $$\sqrt 2$$ and $$2\sqrt 2$$ respective... Match the following : $$(3, 0)$$ is the pt. from which three normals are drawn to the parabola $${y^2} = 4x$$ which meet the parabola in the points $$... The minimum area of triangle formed by the tangent to the$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$and coordinate axes is Tangent to the curve$$y = {x^2} + 6$$at a point$$(1, 7)$$touches the circle$${x^2} + {y^2} + 16x + 12y + c = 0$$at a point$$Q$$. Then the coord... The angle between the tangents drawn from the point$$(1, 4)$$to the parabola$${y^2} = 4x$$is If tangents are drawn to the ellipse$${x^2} + 2{y^2} = 2,$$then the locus of the mid-point of the intercept made by the tangents between the coordin... If the line$$62x + \sqrt 6 y = 2$$touches the hyperbola$${x^2} - 2{y^2} = 4$$, then the point of contact is The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse$${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1,$$i... For hyperbola$${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$which of the following remains constant with change ... The focal chord to$${y^2} = 16x$$is tangent to$${\left( {x - 6} \right)^2} + {y^2} = 2,$$then the possible values of the slope of the chord, are If$$a > 2b > 0$$then the positive value of$$m$$for which$$y = mx - b\sqrt {1 + {m^2}} $$is a common tangent to$${x^2} + {y^2} = {b^2}$$a... The equation of the common tangent to the curves$${y^2} = 8x$$and$$xy = - 1$$is The locus of the mid-point of the line segment joining the focus to a moving point on the parabola$${y^2} = 4ax$$is another parabola with directrix The equation of the directrix of the parabola$${y^2} + 4y + 4x + 2 = 0$$The equation of the common tangent touching the circle$${\left( {x - 3} \right)^2} + {y^2} = 9$$and the parabola$${y^2} = 4x$$above the$$x$$-axis... If$$x + y = k$$is normal to$${y^2} = 12x,$$then$$k$$is If the line$$x - 1 = 0$$is the directrix of the parabola$${y^2} - kx + 8 = 0,$$then one of the values of$$k$$is Let$$P\left( {a\,\sec \,\theta ,\,\,b\,\tan \theta } \right)$$and$$Q\left( {a\,\sec \,\,\phi ,\,\,b\,\tan \,\phi } \right)$$, where$$\t... The curve described parametrically by $$x = {t^2} + t + 1,$$ $$y = {t^2} - t + 1$$ represents If $$x$$ $$=$$ $$9$$ is the chord of contact of the hyperbola $${x^2} - {y^2} = 9,$$ then the equation of the vcorresponding pair of tangents is The number of values of $$c$$ such that the straight line $$y=4x + c$$ touches the curve $$\left( {{x^2}/4} \right) + {y^2} = 1$$ is If $$P=(x, y)$$, $${F_1} = \left( {3,0} \right),\,{F_2} = \left( { - 3,0} \right)$$ and $$16{x^2} + 25{y^2} = 400,$$ then $$P{F_1} + P{F_2}$$ equals Consider a circle with its centre lying on the focus of the parabola $${y^2} = 2px$$ such that it touches the directrix of the parabola. Then a point ... The radius of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and having its centre at $$(0, 3)$$... The equation $$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$ represents Let $$E$$ be the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ and $$C$$ be the circle $${x^2} + {y^2} = 9$$. Let $$P$$ and $$Q$$ be the point... The equation $${{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1$$ represents Each of the four inequalties given below defines a region in the $$xy$$ plane. One of these four regions does not have the following property. For any... ## Subjective Match the conics in Column $$I$$ with the statements/expressions in Column $$II$$. Column $$I$$ (A) Circle (B) Parabola (C) Ellipse (D) Hyperbola ... Match the statements in Column $$I$$ with the properties in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \ti... Tangents are drawn from any point on the hyperbola$${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$to the circle$${x^2} + {y^2} = 9$$.Find the locus of... Find the equation of the common tangent in$${1^{st}}$$quadrant to the circle$${x^2} + {y^2} = 16$$and the ellipse$${{{x^2}} \over {25}} + {{{y^2}... Tangent is drawn to parabola $${y^2} - 2y - 4x + 5 = 0$$ at a point $$P$$ which cuts the directrix at the point $$Q$$. $$A$$ point $$R$$ is such that ... Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If locus of $$P$$ with $${m_1}$$ $$... Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet... Let$$P$$be a point on the ellipse$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 < b < a$$. Let the line parallel to$$y$$-axis pas... Let$$ABC$$be an equilateral triangle inscribed in the circle$${x^2} + {y^2} = {a^2}$$. Suppose perpendiculars from$$A, B, C$$to the major axis of... Let$${C_1}$$and$${C_2}$$be respectively, the parabolas$${x^2} = y - 1$$and$${y^2} = x - 1$$. Let$$P$$be any point on$${C_1}$$and$$Q$$be a... Find the co-ordinates of all the points$$P$$on the ellipse$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, for which the area of the tria... Consider the family of circles$${x^2} + {y^2} = {r^2},\,\,2 < r < 5$$. If in the first quadrant, the common taingent to a circle of this family... The angle between a pair of tangents drawn from a point$$P$$to the parabola$${y^2} = 4ax$$is$${45^ \circ }$$. Show that the locus of the point$$... A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x2 + 2y2 = 6 are at... Points $$A, B$$ and $$C$$ lie on the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$A, B$$ and $$C$$, taken in pairs, intersect at points... From a point $$A$$ common tangents are drawn to the circle $${x^2} + {y^2} = {a^2}/2$$ and parabola $${y^2} = 4ax$$. Find the area of the quadrilater... Show that the locus of a point that divides a chord of slope $$2$$ of the parabola $${y^2} = 4x$$ internally in the ratio $$1:2$$ is a parabola. Find ... Let '$$d$$' be the perpendicular distance from the centre of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ to the tangent draw... Through the vertex $$O$$ of parabola $${y^2} = 4x$$, chords $$OP$$ and $$OQ$$ are drawn at right angles to one another . Show that for all positions o... Three normals are drawn from the point $$(c, 0)$$ to the curve $${y^2} = x.$$ Show that $$c$$ must be greater than $$1/2$$. One normal is always the$...
$$A$$ is point on the parabola $${y^2} = 4ax$$. The normal at $$A$$ cuts the parabola again at point $$B$$. If $$AB$$ subtends a right angle at the ve...
Suppose that the normals drawn at three different points on the parabola $${y^2} = 4x$$ pass through the point $$(h, k)$$. Show that $$h>2$$.

## Fill in the Blanks

An ellipse has eccentricity $${1 \over 2}$$ and one focus at the point $$P\left( {{1 \over 2},1} \right)$$. Its one directrix is the common tangent, n...
The point of intersection of the tangents at the ends of the latus rectum of the parabola $${y^2} = 4x$$ is ...... .
EXAM MAP
Joint Entrance Examination