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JEE Advanced

Conic Sections

Mathematics

Previous Years Questions

Numerical

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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)

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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 > 1 and b < 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} < 0,{y_2} < 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)

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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 > b > 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} < 0}$$ and $${{x_2} > 0}$...
JEE Advanced 2016 Paper 2 Offline
Let $${F_1}\left( {{x_1},0} \right)$$ and $${F_2}\left( {{x_2},0} \right)$$ for $${{x_1} < 0}$$ and $${{x_2} > 0}$...
JEE Advanced 2016 Paper 2 Offline
Let $$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 $$a > 2b > 0$$ 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 > 1$$ represents
IIT-JEE 1981

Subjective

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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 $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 < b < a$$. 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 < r < 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

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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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