Parabola — JEE Main PYQs

Let O be the vertex of the parabola $x^2=4y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C, which is bisected at the point $(1,2)$, is:

Let $y^2=12x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $\angle OPA=90^{\circ }$. Then the locus of the centroid of such triangles OPA is :

Let one end of a focal chord of the parabola $y^2=16x$ be (16, 16). If $P(\alpha ,\beta )$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha +\beta$ is equal to :

If the chord joining the points ${\mathrm{P}}_1\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ and ${\mathrm{P}}_2\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ on the parabola $y^2=12x$ subtends a right angle at the vertex of the parabola, then $x_1{x}_2-y_1{y}_2$ is equal t …

Let the locus of the mid-point of the chord through the origin O of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio $3:1$, is :

An equilateral triangle OAB is inscribed in the parabola ${\mathrm{y}}^2=4\mathrm{x}$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

Let the image of parabola $x^2=4y$, in the line $x-y=1$ be $(\mathrm{y}+\alpha {)}^2=\mathrm{b}(\mathrm{x}-\mathrm{c}),\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{N}$. Then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to

Let A be the focus of the parabola ${\mathrm{y}}^2=8\mathrm{x}$. Let the line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ intersect the parabola at two distinct points B and C . If the centroid of the triangle ABC is $\left(\frac{7}{3},\frac{4}{3}\right)$, then …

Let $\mathrm{A},\mathrm{B}$ and C be the vertices of a variable right angled triangle inscribed in the parabola ${\mathrm{y}}^2=16\mathrm{x}$. Let the vertex B containing the right angle be $(4,8)$ and the locus of the centroid of $△\mathrm{ABC}$ be a conic …

Let the directrix of the parabola $P:y^2=8x$, cut x -axis at the point A . Let $\mathrm{B}(\alpha ,\beta ),\alpha >1$, be a point on P such that the slope of AB is $3/5$. If BC is a focal chord of P , then six times the area of $△\mathrm{ABC}$ is:

Let chord PQ of length $3\sqrt{13}$ of the parabola $y^2=12x$ be such that the ordinates of points P and Q are in the ratio $1:2$. If the chord PQ subtends an angle $\alpha$ at the focus of the parabola, then $\sin \alpha$ is equal to:

Let O be the vertex of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is :

Let $\mathrm{P}(4,4\sqrt{3})$ be a point on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is …

If the line $3x-2y+12=0$ intersects the parabola $4\mathrm{y}=3{\mathrm{x}}^2$ at the points A and B , then at the vertex of the parabola, the line segment AB subtends an angle equal to

Let the shortest distance from $(a,0)$, $a>0$, to the parabola $y^2=4x$ be 4. Then the equation of the circle passing through the point $(a,0)$ and the focus of the parabola, and having its centre on the axis of the parabola is:

The focus of the parabola $y^2=4x+16$ is the centre of the circle C of radius 5. If the values of $\lambda$, for which C passes through the point of intersection of the lines $3x-y=0$ and $x+\lambda y=4$, are ${\lambda }_1$ and ${\lambda }_2$, ${\lambda }_1<{\lambda }_2$, then …

Let $y^2=12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ)=\frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2+64y^2-\alpha x-64\sqrt{3}y=\beta$, then …

Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersects at the points A and B , then $(\mathrm{AB}{)}^2$ is equal to

Let A and B be the two points of intersection of the line $\mathrm{y}+5=0$ and the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $d$ denotes the distance between A and B , and a denotes the area of $△\mathrm{SAB}$, where S is the focus o …

Let ABCD be a trapezium whose vertices lie on the parabola ${\mathrm{y}}^2=4\mathrm{x}$. Let the sides AD and BC of the trapezium be parallel to y -axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point $(1,0)$, then the area of ABCD is :

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2},3\right)$ and the directrix $\mathrm{x}+2\mathrm{y}=0$ is $\alpha x^2+\beta y^2-\gamma xy-30x-60y+225=0$, then $\alpha +\beta +\gamma$ is equal to:

Let the point P of the focal chord PQ of the parabola $y^2=16x$ be $(1,-4)$. If the focus of the parabola divides the chord PQ in the ratio $m:n$, $\gcd (m,n)=1$, then $m^2+n^2$ is equal to :

The shortest distance between the curves ${\mathrm{y}}^2=8\mathrm{x}$ and $x^2+y^2+12y+35=0$ is :

Let the focal chord PQ of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ make an angle of $60^{\circ }$ with the positive x -axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y -axis at the point …

The radius of the smallest circle which touches the parabolas $\mathrm{y}={\mathrm{x}}^2+2$ and $\mathrm{x}={\mathrm{y}}^2+2$ is

The axis of a parabola is the line $\mathrm{y}=\mathrm{x}$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2\sqrt{2}$ units from the origin, respectively. If the point $(1,\mathrm{k})$ lies on the parabola, then a possible value of k is : …

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2x+y+2=0$. Then the sum of the ordinates of the points on P , whose abscissa is -2 , is

Let the length of the focal chord PQ of the parabola $y^2=12x$ be 15 units. If the distance of PQ from the origin is $p$, then $10p^2$ is equal to ____.

Let $PQ$ be a chord of the parabola $y^2=12x$ and the midpoint of $PQ$ be at $(4,1)$. Then, which of the following points lies on the line passing through the points $P$ and $Q$?

Let a line perpendicular to the line $2x-y=10$ touch the parabola $y^2=4(x-9)$ at the point $P$. The distance of the point $P$ from the centre of the circle $x^2+y^2-14x-8y+56=0$ is ____ .

Let a conic $C$ pass through the point $(4,-2)$ and $P(x,y)$, $x\ge 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of …

Let $\mathrm{A},\mathrm{B}$ and C be three points on the parabola $y^2=6x$ and let the line segment $AB$ meet the line $L$ through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L.

Then …

Let $P(\alpha ,\beta )$ be a point on the parabola $y^2=4x$. If P also lies on the chord of the parabola $x^2=8y$ whose mid point is $\left(1,\frac{5}{4}\right)$. Then $(\alpha -28)(\beta -8)$ is equal to ______.

If the shortest distance of the parabola $y^2=4x$ from the centre of the circle $x^2+y^2-4x-16y+64=0$ is d, then $d^2$ is equal to :

Let R be the focus of the parabola ${\mathrm{y}}^2=20\mathrm{x}$ and the line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ intersect the parabola at two points P and Q. Let the point $\mathrm{G}(10,10)$ be the centroid of the triangle PQR. If $\mathrm{c}-\mathrm{m}=6$, then …

Let $\mathrm{A}(0,1)$, $\mathrm{B}(1,1)$ and $\mathrm{C}(1,0)$ be the mid-points of the sides of a triangle with incentre at the point D. If the focus of the parabola $y^2=4ax$ passing through D is $(\alpha +\beta \sqrt{2},0)$, where $\alpha$ and $\beta$ are rational numbers, …

The ordinates of the points P and Q on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the ratio $3:1$. If $R(\alpha ,\beta )$ is the point of intersection of the tangents to the parabola at P and Q, then $\frac{{\beta }^2}{\alpha }$ is equal to ______.

Let a common tangent to the curves $y^2=4x$ and $(x-4{)}^2+y^2=16$ touch the curves at the points P and Q. Then $(PQ{)}^2$ is equal to ______.

Let PQ be a focal chord of the parabola $y^2=36x$ of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that $\mathrm{PM}:\mathrm{MQ}=3:1$. Then which of the following points does NO …

Let a tangent to the curve ${\mathrm{y}}^2=24\mathrm{x}$ meet the curve $\mathrm{xy}=2$ at the points A and B . Then the mid points of such line segments AB lie on a parabola with the

If $\mathrm{P}(\mathrm{h},\mathrm{k})$ be point on the parabola $\mathrm{x}=4{\mathrm{y}}^2$, which is nearest to the point $\mathrm{Q}(0,33)$, then the distance of P from the directrix of the parabola $y^2=4(x+y)$ is equal to:

If the x-intercept of a focal chord of the parabola $y^2=8x+4y+4$ is 3, then the length of this chord is equal to ______.

The equations of two sides of a variable triangle are $\mathrm{x}=0$ and $\mathrm{y}=3$, and its third side is a tangent to the parabola ${\mathrm{y}}^2=6\mathrm{x}$. The locus of its circumcentre is :

Let A be a point on the x-axis. Common tangents are drawn from A to the curves ${\mathrm{x}}^2+{\mathrm{y}}^2=8$ and ${\mathrm{y}}^2=16x$. If one of these tangents touches the two curves at Q and R, then $(\mathrm{QR}{)}^2$ is equal to

Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2},0\right)$ and directrix $y=-\frac{1}{2}$.

Then
$S=\left\{x\in \mathbb{R}:{\tan }^{-1}\left(\sqrt{f(x)}\right)+{\sin }^{-1}(\sqrt{f(x)+1})=\frac{\pi }{2}\right\}$:

The distance of the point $(6,-2\sqrt{2})$ from the common tangent $\mathrm{y}=\mathrm{mx}+\mathrm{c},\mathrm{m}>0$, of the curves $\mathrm{x}=2{\mathrm{y}}^2$ and $\mathrm{x}=1+{\mathrm{y}}^2$ is

If the tangents drawn at the points P and Q on the parabola ${\mathrm{y}}^2=2\mathrm{x}-3$ intersect at the point $\mathrm{R}(0,1)$, then the orthocentre of the triangle PQR is :

Let the hyperbola $\mathrm{H}:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ pass through the point $(2\sqrt{2},-2\sqrt{2})$. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola …

Two tangent lines $l_1$ and $l_2$ are drawn from the point $(2,0)$ to the parabola $2y^2=-x$. If the lines $l_1$ and $l_2$ are also tangent to the circle $(x-5{)}^2+y^2=r$, then 17 r is equal to

Let $\mathrm{P}(a,\mathrm{b})$ be a point on the parabola ${\mathrm{y}}^2=8\mathrm{x}$ such that the tangent at P passes through the centre of the circle $x^2+y^2-10x-14y+65=0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of al …