Parabola · Mathematics · JEE Advanced
Start PracticeMCQ (More than One Correct Answer)
JEE Advanced 2024 Paper 2 Online
Let $A_1, B_1, C_1$ be three points in the $x y$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8 x$ at $A_1$ an...
JEE Advanced 2022 Paper 1 Online
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...
JEE Advanced 2021 Paper 2 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 PQ and PQ' are tangents to E....
JEE Advanced 2017 Paper 1 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 which of the following is(are) po...
JEE Advanced 2016 Paper 2 Offline
Let $$P$$ be the point on the parabola $${y^2} = 4x$$ which is at the shortest distance from the center $$S$$ of the circle $${x^2} + {y^2} - 4x - 16y...
JEE Advanced 2016 Paper 1 Offline
The circle $${C_1}:{x^2} + {y^2} = 3,$$ with centre at $$O$$, intersects the parabola $${x^2} = 2y$$ at the point $$P$$ in the first quadrant, Let the...
JEE Advanced 2015 Paper 1 Offline
Let $$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$$ of th...
IIT-JEE 2011 Paper 2 Offline
Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by
IIT-JEE 2010 Paper 1 Offline
Let $$A$$ and $$B$$ be two distinct points on the parabola $${y^2} = 4x$$. If the axis of the parabola touches a circle of radius $$r$$ having $$AB$$ ...
IIT-JEE 2009 Paper 2 Offline
The tangent $$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$$, respectively....
IIT-JEE 2006
The equations of the common tangents to the parabola $$y = {x^2}$$ and $$y = - {\left( {x - 2} \right)^2}$$ is/are
Numerical
JEE Advanced 2024 Paper 2 Online
A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha)$ to the parabola $x^2=-4 a y$, where $a>0$. Let $L$ be the line passing ...
JEE Advanced 2015 Paper 2 Offline
Suppose that the foci of the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1$$ are $$\left( {{f_1},0} \right)$$ and $$\left( {{f_2},0} \right)$$ w...
JEE Advanced 2015 Paper 1 Offline
If the normals of the parabola $${y^2} = 4x$$ drawn at the end points of its latus rectum are tangents to the circle $${\left( {x - 3} \right)^2} + {\...
JEE Advanced 2015 Paper 1 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 $$A$$ and $$B$$ are the points of inte...
IIT-JEE 2012 Paper 1 Offline
Let $$S$$ be the focus of the parabola $${y^2} = 8x$$ and let $$PQ$$ be the common chord of the circle $${x^2} + {y^2} - 2x - 4y = 0$$ and the given p...
IIT-JEE 2011 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 latus rectum and the point $$P\l...
MCQ (Single Correct Answer)
JEE Advanced 2023 Paper 1 Online
Let $P$ be a point on the parabola $y^2=4 a x$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the tria...
JEE Advanced 2020 Paper 1 Offline
Let a, b and $$\lambda $$ be positive real numbers. Suppose P is an end point of the latus return of the parabola y2 = 4$$\lambda $$x, and suppose the...
JEE Advanced 2019 Paper 2 Offline
Let the circles C1 : x2 + y2 = 9 and C2 : (x $$-$$ 3)2 + (y $$-$$ 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x $$-$$...
JEE Advanced 2019 Paper 2 Offline
Let the circle C1 : x2 + y2 = 9 and C2 : (x $$-$$ 3)2 + (y $$-$$ 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x $$-$$ ...
JEE Advanced 2017 Paper 1 Offline
If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the o...
JEE Advanced 2014 Paper 2 Offline
Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$ and $$S\,\,\left( {a...
JEE Advanced 2014 Paper 2 Offline
Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$ and $$S\,\,\left( {a...
JEE Advanced 2013 Paper 2 Offline
A line $$L:y=mx+3$$ meets $$y$$-axis at R$$(0, 3)$$ and the arc of the parabola $${y^2} = 16x,$$ $$0 \le y \le 6$$ at the point $$F\left( {{x_0},{y_0}...
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 point lying on the line $$y=2x+...
JEE Advanced 2013 Paper 2 Offline
Let $$PQ$$ be a focal chord of the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$P$$ and $$Q$$ meet at a point lying on the line $$y=2x+...
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 $$(0, 0)$$ to $$(x, y)$$ in the ...
IIT-JEE 2009 Paper 2 Offline
The locus of the orthocentre of the triangle formed by the lines $$(1 + p)x - py + p(1 + p) = 0,
$$
$$(1 + q)x - qy + q(1 + q) = 0$$
and $$y = 0$$, w...
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 first and the fourth quadrants, re...
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 first and the fourth quadrants, re...
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 first and the fourth quadrants, re...
IIT-JEE 2007
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...
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 $$\sqrt 2 $$ and $$2\sqrt 2 $$ respective...
IIT-JEE 2006
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 $$...
IIT-JEE 2005 Screening
Tangent to the curve $$y = {x^2} + 6$$ at a point $$(1, 7)$$ touches the circle $${x^2} + {y^2} + 16x + 12y + c = 0$$ at a point $$Q$$. Then the coord...
IIT-JEE 2004 Screening
The angle between the tangents drawn from the point $$(1, 4)$$ to the parabola $${y^2} = 4x$$ is
IIT-JEE 2003 Screening
The focal chord to $${y^2} = 16x$$ is tangent to $${\left( {x - 6} \right)^2} + {y^2} = 2,$$ then the possible values of the slope of the chord, are
IIT-JEE 2002 Screening
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix
IIT-JEE 2002 Screening
The equation of the common tangent to the curves $${y^2} = 8x$$ and $$xy = - 1$$ is
IIT-JEE 2001 Screening
The equation of the directrix of the parabola $${y^2} + 4y + 4x + 2 = 0$$
IIT-JEE 2001 Screening
The equation of the common tangent touching the circle $${\left( {x - 3} \right)^2} + {y^2} = 9$$ and the parabola $${y^2} = 4x$$ above the $$x$$-axis...
IIT-JEE 2000 Screening
If $$x + y = k$$ is normal to $${y^2} = 12x,$$ then $$k$$ is
IIT-JEE 2000 Screening
If the line $$x - 1 = 0$$ is the directrix of the parabola $${y^2} - kx + 8 = 0,$$ then one of the values of $$k$$ is
IIT-JEE 1999
The curve described parametrically by $$x = {t^2} + t + 1,$$ $$y = {t^2} - t + 1 $$ represents
IIT-JEE 1995 Screening
Consider a circle with its centre lying on the focus of the parabola $${y^2} = 2px$$ such that it touches the directrix of the parabola. Then a point ...
Subjective
IIT-JEE 2004
Tangent is drawn to parabola $${y^2} - 2y - 4x + 5 = 0$$ at a point $$P$$ which cuts the directrix at the point $$Q$$. $$A$$ point $$R$$ is such that ...
IIT-JEE 2003
Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If locus of $$P$$ with $${m_1}$$ $$...
IIT-JEE 2000
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...
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 in pairs, intersect at points...
IIT-JEE 1996
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...
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 ratio $$1:2$$ is a parabola. Find ...
IIT-JEE 1994
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...
IIT-JEE 1991
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 $...
IIT-JEE 1982
$$A$$ 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...
IIT-JEE 1981
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
IIT-JEE 1994
The point of intersection of the tangents at the ends of the latus rectum of the parabola $${y^2} = 4x$$ is ...... .