NEW
New Website Launch
Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc...

## Numerical

More
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

## MCQ (More than One Correct Answer)

More
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$$then which of... JEE Advanced 2017 Paper 1 Offline Let$$P$$be the point on the parabola$${y^2} = 4x$$which is at the shortest distance from the center$$S$$of the cir... JEE Advanced 2016 Paper 2 Offline The circle$${C_1}:{x^2} + {y^2} = 3,$$with centre at$$O$$, intersects the parabola$${x^2} = 2y$$at the point$$P$$... JEE Advanced 2016 Paper 1 Offline Consider the hyperbola$$H:{x^2} - {y^2} = 1$$and a circle$$S$$with center$$N\left( {{x_2},0} \right)$$. Suppose tha... JEE Advanced 2015 Paper 2 Offline Let$$P$$and$$Q$$be distinct points on the parabola$${y^2} = 2x$$such that a circle with$$PQ$$as diameter passes ... JEE Advanced 2015 Paper 1 Offline Let$${E_1}$$and$${E_2}$$be two ellipses whose centres are at the origin. The major axes of$${E_1}$$and$${E_2}$$l... JEE Advanced 2015 Paper 2 Offline Tangents are drawn to the hyperbola$${{{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
In a triangle $$ABC$$ with fixed base $$BC$$, the vertex $$A$$ moves such that $$\cos \,B + \cos \,C = 4{\sin ^2}{A \o... IIT-JEE 2009 The tangent$$PT$$and the normal$$PN$$to the parabola$${y^2} = 4ax$$at a point$$P$$on it meet its axis at points ... IIT-JEE 2009 Let$$P\left( {{x_1},{y_1}} \right)$$and$$Q\left( {{x_2},{y_2}} \right),{y_1} &lt; 0,{y_2} &lt; 0,$$be the end points... IIT-JEE 2008 The equations of the common tangents to the parabola$$y = {x^2}$$and$$y = - {\left( {x - 2} \right)^2}$$is/are IIT-JEE 2006 Let a hyperbola passes through the focus of the ellipse$${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$. The transve... IIT-JEE 2006 On the ellipse$$4{x^2} + 9{y^2} = 1,$$the points at which the tangents are parallel to the line$$8x = 9y$$are IIT-JEE 1999 ## MCQ (Single Correct Answer) More Let a, b and$$\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 $${{{x^2}} \over 9... IIT-JEE 2003 Screening The locus of the mid-point of the line segment joining the focus to a moving point on the parabola$${y^2} = 4ax$$is an... IIT-JEE 2002 Screening The equation of the common tangent to the curves$${y^2} = 8x$$and$$xy = - 1$$is IIT-JEE 2002 Screening If$$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 ## 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 $$... IIT-JEE 2000 Consider the family of circles$${x^2} + {y^2} = {r^2},\,\,2 &lt; r &lt; 5$$. If in the first quadrant, the common taing... IIT-JEE 1999 Find the co-ordinates of all the points$$P$$on the ellipse$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, ... IIT-JEE 1999 The angle between a pair of tangents drawn from a point$$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

More
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

### Joint Entrance Examination

JEE Main JEE Advanced WB JEE

### Graduate Aptitude Test in Engineering

GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN

NEET

Class 12