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## Numerical

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Consider the hyperbola $$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$$ with foci at $S$ and $S_{1}$, where $S$ lies on the ...
JEE Advanced 2022 Paper 2 Online
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, l...
JEE Advanced 2021 Paper 2 Online
For how many values of p, the circle x2 + y2 + 2x + 4y $$-$$ p = 0 and the coordinate axes have exactly three common poi...
JEE Advanced 2017 Paper 1 Offline
Suppose that the foci of the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1$$ are $$\left( {{f_1},0} \right)$$ and ...
JEE Advanced 2015 Paper 2 Offline
Let the curve $$C$$ be the mirror image of the parabola $${y^2} = 4x$$ with respect to the line $$x+y+4=0$$. If $$A$$ an...
JEE Advanced 2015 Paper 1 Offline
If the normals of the parabola $${y^2} = 4x$$ drawn at the end points of its latus rectum are tangents to the circle $${... JEE Advanced 2015 Paper 1 Offline A vertical line passing through the point$$(h,0)$$intersects the ellipse$${{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$$... JEE Advanced 2013 Paper 1 Offline Let$$S$$be the focus of the parabola$${y^2} = 8x$$and let$$PQ$$be the common chord of the circle$${x^2} + {y^2} -...
IIT-JEE 2012 Paper 1 Offline
Consider the parabola $${y^2} = 8x$$. Let $${\Delta _1}$$ be the area of the triangle formed by the end points of its la...
IIT-JEE 2011 Paper 1 Offline
The line $$2x + y = 1$$ is tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If this l...
IIT-JEE 2010 Paper 1 Offline

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Consider the ellipse $$\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$$$Let$H(\alpha, 0), 0 .tg {border-collapse:collapse;bo...
JEE Advanced 2022 Paper 1 Online
Let a, b and $$\lambda$$ be positive real numbers. Suppose P is an end point of the latus return of the parabola y2 = 4...
JEE Advanced 2020 Paper 1 Offline
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...
JEE Advanced 2019 Paper 2 Offline
Let the circles C1 : x2 + y2 = 9 and C2 : (x $$-$$ 3)2 + (y $$-$$ 4)2 = 16, intersect at the points X and Y. Suppose tha...
JEE Advanced 2019 Paper 2 Offline
Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, where a &gt; b &gt; 0, be a hyperbola in the XY-plane w...
JEE Advanced 2018 Paper 2 Offline
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 coordina...
JEE Advanced 2018 Paper 1 Offline
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 throug...
JEE Advanced 2018 Paper 1 Offline
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 ...
JEE Advanced 2017 Paper 1 Offline
The tangent to a suitable conic (Column 1) at $$\left( {\sqrt 3 ,\,{1 \over 2}} \right)$$ is found to be $$\sqrt 3 x + 2... JEE Advanced 2017 Paper 1 Offline For$$a = \sqrt 2 $$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact ($$-$$1, 1), then whi... JEE Advanced 2017 Paper 1 Offline Let$${F_1}\left( {{x_1},0} \right)$$and$${F_2}\left( {{x_2},0} \right)$$for$${{x_1} &lt; 0}$$and$${{x_2} &gt; 0}$... JEE Advanced 2016 Paper 2 Offline Let $${F_1}\left( {{x_1},0} \right)$$ and $${F_2}\left( {{x_2},0} \right)$$ for $${{x_1} &lt; 0}$$ and $${{x_2} &gt; 0}... JEE Advanced 2016 Paper 2 Offline Let$$a, r, s, t$$be nonzero real numbers. Let$$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} ... JEE Advanced 2014 Paper 2 Offline Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} ... JEE Advanced 2014 Paper 2 Offline The common tangents to the circle$${x^2} + {y^2} = 2$$and the parabola$${y^2} = 8x$$touch the circle at the points ... JEE Advanced 2014 Paper 2 Offline Let$$PQ$$be a focal chord of the parabola$${y^2} = 4ax$$. The tangents to the parabola at$$P$$and$$Q$$meet at a p... JEE Advanced 2013 Paper 2 Offline Let$$PQ$$be a focal chord of the parabola$${y^2} = 4ax$$. The tangents to the parabola at$$P$$and$$Q$$meet at a p... JEE Advanced 2013 Paper 2 Offline A line$$L:y=mx+3$$meets$$y$$-axis at$$(E, 3)$$and the are of the parabola$${y^2} = 16x,0 \le y \le 6$$at the... JEE Advanced 2013 Paper 2 Offline The ellipse$${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$is inscribed in a rectangle$$R$$whose sides are parall... IIT-JEE 2012 Paper 1 Offline Let$$P(6, 3)$$be a point on the hyperbola$${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal at ... IIT-JEE 2011 Paper 2 Offline Let$$(x, y)$$be any point on the parabola$${y^2} = 4x$$. Let$$P$$be the point that divides the line segment from$$... IIT-JEE 2011 Paper 2 Offline Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ... IIT-JEE 2010 Paper 2 Offline Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ... IIT-JEE 2010 Paper 2 Offline Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ... IIT-JEE 2010 Paper 2 Offline The circle $${x^2} + {y^2} - 8x = 0$$ and hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ intersect at the point... IIT-JEE 2010 Paper 1 Offline The circle $${x^2} + {y^2} - 8x = 0$$ and hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ intersect at the point... IIT-JEE 2010 Paper 1 Offline The locus of the orthocentre of the triangle formed by the lines $$\left( {1 + p} \right)x - py + p\left( {1 + p} \rig... IIT-JEE 2009 The normal at a point$$P$$on the ellipse$${x^2} + 4{y^2} = 16$$meets the$$x$$- axis$$Q$$. If$$M$$is the mid poin... IIT-JEE 2009 The line passing through the extremity$$A$$of the major axis and extremity$$B$$of the minor axis of the ellipse$${x... IIT-JEE 2009 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 $$A... IIT-JEE 2008 Let$$a$$and$$b$$be non-zero real numbers. Then, the equation$${x^2}{\cos ^2}\theta - {y^2}{\sin ^2}\theta = 0$$r... IIT-JEE 2008 STATEMENT-1: The curve$$y = {{ - {x^2}} \over 2} + x + 1$$is symmetric with respect to the line$$x=1$$. because STATE... IIT-JEE 2007 Consider the circle$${x^2} + {y^2} = 9$$and the parabola$${y^2} = 8x$$. They intersect at$$P$$and$$Q$$in the firs... IIT-JEE 2007 Consider the circle$${x^2} + {y^2} = 9$$and the parabola$${y^2} = 8x$$. They intersect at$$P$$and$$Q$$in the firs... IIT-JEE 2007 Consider the circle$${x^2} + {y^2} = 9$$and the parabola$${y^2} = 8x$$. They intersect at$$P$$and$$Q$$in the firs... IIT-JEE 2007 A hyperbola, having the transverse axis of length$$2\sin \theta ,$$is confocal with the ellipse$$3{x^2} + 4{y^2} = 12...
IIT-JEE 2007
Match the following : $$(3, 0)$$ is the pt. from which three normals are drawn to the parabola $${y^2} = 4x$$ which meet...
IIT-JEE 2006
The axis of a parabola is along the line $$y = x$$ and the distances of its vertex and focus from origin are $$\sqrt 2 ... IIT-JEE 2006 Tangent to the curve$$y = {x^2} + 6$$at a point$$(1, 7)$$touches the circle$${x^2} + {y^2} + 16x + 12y + c = 0$$at... IIT-JEE 2005 Screening The minimum area of triangle formed by the tangent to the$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$and ... IIT-JEE 2005 Screening If the line$$62x + \sqrt 6 y = 2$$touches the hyperbola$${x^2} - 2{y^2} = 4$$, then the point of contact is IIT-JEE 2004 Screening If tangents are drawn to the ellipse$${x^2} + 2{y^2} = 2,$$then the locus of the mid-point of the intercept made by th... IIT-JEE 2004 Screening The angle between the tangents drawn from the point$$(1, 4)$$to the parabola$${y^2} = 4x$$is IIT-JEE 2004 Screening The focal chord to$${y^2} = 16x$$is tangent to$${\left( {x - 6} \right)^2} + {y^2} = 2,$$then the possible values of... IIT-JEE 2003 Screening For hyperbola$${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$which of the following... IIT-JEE 2003 Screening The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse$${{{x^2}} \over 9...
IIT-JEE 2003 Screening
The equation of the common tangent to the curves $${y^2} = 8x$$ and $$xy = - 1$$ is
IIT-JEE 2002 Screening
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is an...
IIT-JEE 2002 Screening
If $$a &gt; 2b &gt; 0$$ then the positive value of $$m$$ for which $$y = mx - b\sqrt {1 + {m^2}}$$ is a common tangent ...
IIT-JEE 2002 Screening
The equation of the common tangent touching the circle $${\left( {x - 3} \right)^2} + {y^2} = 9$$ and the parabola $${y^... IIT-JEE 2001 Screening The equation of the directrix of the parabola$${y^2} + 4y + 4x + 2 = 0$$IIT-JEE 2001 Screening If the line$$x - 1 = 0$$is the directrix of the parabola$${y^2} - kx + 8 = 0,$$then one of the values of$$k$$is IIT-JEE 2000 Screening If$$x + y = k$$is normal to$${y^2} = 12x,$$then$$k$$is IIT-JEE 2000 Screening If$$x=9$$is the chord of contact of the hyperbola$${x^2} - {y^2} = 9,$$then the equation of the vcorrespon... IIT-JEE 1999 The curve described parametrically by$$x = {t^2} + t + 1,y = {t^2} - t + 1 $$represents IIT-JEE 1999 Let$$P\left( {a\,\sec \,\theta ,\,\,b\,\tan \theta } \right)$$and$$Q\left( {a\,\sec \,\,\phi ,\,\,b\,\tan ...
IIT-JEE 1999
If $$P=(x, y)$$, $${F_1} = \left( {3,0} \right),\,{F_2} = \left( { - 3,0} \right)$$ and $$16{x^2} + 25{y^2} = 400,$$ the...
IIT-JEE 1998
The number of values of $$c$$ such that the straight line $$y=4x + c$$ touches the curve $$\left( {{x^2}/4} \right) + {... IIT-JEE 1998 The radius of the circle passing through the foci of the ellipse$${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and h... IIT-JEE 1995 Screening Consider a circle with its centre lying on the focus of the parabola$${y^2} = 2px$$such that it touches the directrix ... IIT-JEE 1995 Screening Let$$E$$be the ellipse$${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$and$$C$$be the circle$${x^2} + {y^2} = 9$$. Le... IIT-JEE 1994 The equation$$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$represents IIT-JEE 1994 Each of the four inequalties given below defines a region in the$$xy$$plane. One of these four regions does not have t... IIT-JEE 1981 The equation$${{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r &gt; 1$$represents IIT-JEE 1981 ## MCQ (More than One Correct Answer) More Consider the parabola$$y^{2}=4 x$$. Let$$S$$be the focus of the parabola. A pair of tangents drawn to the parabola fr... JEE Advanced 2022 Paper 1 Online Let E denote the parabola y2 = 8x. Let P = ($$-$$2, 4), and let Q and Q' be two distinct points on E such that the lines... JEE Advanced 2021 Paper 2 Online Let a and b be positive real numbers such that a &gt; 1 and b &lt; a. Let P be a point in the first quadrant that lies o... JEE Advanced 2020 Paper 2 Offline Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :$${E_1}:{{{x^2}} \o...
JEE Advanced 2019 Paper 1 Offline
Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let t...
JEE Advanced 2018 Paper 2 Offline
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...
JEE Advanced 2018 Paper 2 Offline
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 whi...
JEE Advanced 2017 Paper 1 Offline
If $$2x - y + 1 = 0$$ is a tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {16}} = 1$$ then which of...
JEE Advanced 2017 Paper 1 Offline
The circle $${C_1}:{x^2} + {y^2} = 3,$$ with centre at $$O$$, intersects the parabola $${x^2} = 2y$$ at the point $$P$$ ...
JEE Advanced 2016 Paper 1 Offline
Let $$P$$ be the point on the parabola $${y^2} = 4x$$ which is at the shortest distance from the center $$S$$ of the cir...
JEE Advanced 2016 Paper 2 Offline
Consider the hyperbola $$H:{x^2} - {y^2} = 1$$ and a circle $$S$$ with center $$N\left( {{x_2},0} \right)$$. Suppose tha...
JEE Advanced 2015 Paper 2 Offline
Let $${E_1}$$ and $${E_2}$$ be two ellipses whose centres are at the origin. The major axes of $${E_1}$$ and $${E_2}$$ l...
JEE Advanced 2015 Paper 2 Offline
Let $$P$$ and $$Q$$ be distinct points on the parabola $${y^2} = 2x$$ such that a circle with $$PQ$$ as diameter passes ...
JEE Advanced 2015 Paper 1 Offline
Tangents are drawn to the hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1,$$ parallel to the straight line $$2x - ... IIT-JEE 2012 Paper 1 Offline Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by IIT-JEE 2011 Paper 2 Offline Let the eccentricity of the hyperbola$${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$be reciprocal to that of... IIT-JEE 2011 Paper 1 Offline Let$$A$$and$$B$$be two distinct points on the parabola$${y^2} = 4x$$. If the axis of the parabola touches a circle ... IIT-JEE 2010 Paper 1 Offline An ellipse intersects the hyperbola$$2{x^2} - 2{y^2} = 1$$orthogonally. The eccentricity of the ellipse is reciprocal ... IIT-JEE 2009 The tangent$$PT$$and the normal$$PN$$to the parabola$${y^2} = 4ax$$at a point$$P$$on it meet its axis at points ... IIT-JEE 2009 In a triangle$$ABC$$with fixed base$$BC$$, the vertex$$A$$moves such that$$$\cos \,B + \cos \,C = 4{\sin ^2}{A \o... IIT-JEE 2009 Let $$P\left( {{x_1},{y_1}} \right)$$ and $$Q\left( {{x_2},{y_2}} \right),{y_1} &lt; 0,{y_2} &lt; 0,$$ be the end points... IIT-JEE 2008 Let a hyperbola passes through the focus of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$. The transve... IIT-JEE 2006 The equations of the common tangents to the parabola $$y = {x^2}$$ and $$y = - {\left( {x - 2} \right)^2}$$ is/are IIT-JEE 2006 On the ellipse $$4{x^2} + 9{y^2} = 1,$$ the points at which the tangents are parallel to the line $$8x = 9y$$ are IIT-JEE 1999 ## Subjective More Match the conics in Column $$I$$ with the statements/expressions in Column $$II$$. Column $$I$$ (A) Circle (B) Parab... IIT-JEE 2009 Match the statements in Column $$I$$ with the properties in Column $$II$$ and indicate your answer by darkening the appr... IIT-JEE 2007 Find the equation of the common tangent in $${1^{st}}$$ quadrant to the circle $${x^2} + {y^2} = 16$$ and the ellipse $$... IIT-JEE 2005 Tangents are drawn from any point on the hyperbola$${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$to the circle$${x^2} +... IIT-JEE 2005 Tangent is drawn to parabola $${y^2} - 2y - 4x + 5 = 0$$ at a point $$P$$ which cuts the directrix at the point $$Q$$.$...
IIT-JEE 2004
Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If lo...
IIT-JEE 2003
Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellips...
IIT-JEE 2002
Let $$P$$ be a point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 &lt; b &lt; a$$. Let the l...
IIT-JEE 2001
Let $${C_1}$$ and $${C_2}$$ be respectively, the parabolas $${x^2} = y - 1$$ and $${y^2} = x - 1$$. Let $$P$$ be any poi...
IIT-JEE 2000
Let $$ABC$$ be an equilateral triangle inscribed in the circle $${x^2} + {y^2} = {a^2}$$. Suppose perpendiculars from $$... IIT-JEE 2000 Find the co-ordinates of all the points$$P$$on the ellipse$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, ... IIT-JEE 1999 Consider the family of circles$${x^2} + {y^2} = {r^2},\,\,2 &lt; r &lt; 5$$. If in the first quadrant, the common taing... IIT-JEE 1999 The angle between a pair of tangents drawn from a point$$P$$to the parabola$${y^2} = 4ax$$is$${45^ \circ }$$. Show ... IIT-JEE 1998 A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q of t... IIT-JEE 1997 From a point$$A$$common tangents are drawn to the circle$${x^2} + {y^2} = {a^2}/2$$and parabola$${y^2} = 4ax$$. Fin... IIT-JEE 1996 Points$$A, B$$and$$C$$lie on the parabola$${y^2} = 4ax$$. The tangents to the parabola at$$A, B$$and$$C$$, taken... IIT-JEE 1996 Let '$$d$$' be the perpendicular distance from the centre of the ellipse$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^...
IIT-JEE 1995
Show that the locus of a point that divides a chord of slope $$2$$ of the parabola $${y^2} = 4x$$ internally in the rati...
IIT-JEE 1995
Through the vertex $$O$$ of parabola $${y^2} = 4x$$, chords $$OP$$ and $$OQ$$ are drawn at right angles to one another ....
IIT-JEE 1994
Three normals are drawn from the point $$(c, 0)$$ to the curve $${y^2} = x.$$ Show that $$c$$ must be greater than $$1/2... IIT-JEE 1991$$A$$is point on the parabola$${y^2} = 4ax$$. The normal at$$A$$cuts the parabola again at point$$B$$. If$$AB$$su... IIT-JEE 1982 Suppose that the normals drawn at three different points on the parabola$${y^2} = 4x$$pass through the point$$(h, k)\$...
IIT-JEE 1981

## Fill in the Blanks

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An ellipse has eccentricity $${1 \over 2}$$ and one focus at the point $$P\left( {{1 \over 2},1} \right)$$. Its one dire...
IIT-JEE 1996
The point of intersection of the tangents at the ends of the latus rectum of the parabola $${y^2} = 4x$$ is ...... .
IIT-JEE 1994

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