Parabola PYQs - Last 5 Years

Let O be the vertex of the parabola $y^2=4 x$ 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 one root of the quadratic equation in $x$ :
\(\left(k^2-15 k+27\right) x^2+9(k-1) x+18=0\)
be twice the other. Then the length of the latus rectum of the parabola $y^2=6 k x$ is equal to:

Let chord PQ of length $3 \sqrt{13}$ of the parabola $y^2=12 x$ 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 the directrix of the parabola $\mathrm{P}: y^2=8 x$, cut $x$-axis at the point A . Let $\mathrm{B}(\alpha, \beta), \alpha>1$, be a point on $P$ such that the slope of $A B$ is $3 / 5$. If $B C$ is a focal chord of $P$, then six times th...

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

Let the parabola $y = x^2 + px + q$ passing through the point $(1, -1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2 + q^2$ is :

Let A be the focus of the parabola $y^2 = 8x$. Let the line $y = mx + c$ 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 $(BC)^2$ is equal to :

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

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

If the chord joining the points $\mathrm{P}_1\left(x_1, y_1\right)$ and $\mathrm{P}_2\left(x_2, y_2\right)$ on the parabola $y^2=12 x$ subtends a right angle at the vertex of the parabola, then $x_1 x_2-y_1 y_2$ is equal to

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^2=4 x$ 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 :

Let O be the vertex of the parabola $x^2=4 y$ 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 bisecte...

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:

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 $r$ be the radius of the circle, which touches $x$ - axis at point $(a, 0), a<0$ and the parabola $\mathrm{y}^2=9 x$ at the point $(4,6)$. Then $r$ is equal to ______.

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

The axis of a parabola is the line $y=x$ 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 $(1, k)$ lies on the parabola, then a possible value of k...

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}: y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $A B$, parabola $P$ and the $x$-axis, is :

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

Let the focal chord PQ of the parabola $y^2=4 x$ 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 ...

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

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :

Let $y^2=12 x$ be the parabola and $S$ be its focus. Let $P Q$ be a focal chord of the parabola such that $(S P)(S Q)=\frac{147}{4}$. Let $C$ be the circle described taking $P Q$ as a diameter. If the equation of a circle $C$ is $64 x^2+64 ...

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 $(1,0)$...

Let $A$ and $B$ be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola $y^2=4 x$ with respect to the line $x+y+4=0$. If $d$ denotes the distance between $A$ and $B$, and a denotes the area of $\triangle S...

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

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

The focus of the parabola $y^2=4 x+16$ 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 $3 x-y=0$ and $x+\lambda y=4$, are $\lambda_1$ and $\lambda_2...

Let the shortest distance from $(a, 0), a>0$, to the parabola $y^2=4 x$ 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 :

Let the parabola $y=x^2+\mathrm{p} x-3$, meet the coordinate axes at the points $\mathrm{P}, \mathrm{Q}$ and R . If the circle C with centre at $(-1,-1)$ passes through the points $P, Q$ and $R$, then the area of $\triangle P Q R$ is :

Let $\mathrm{P}(4,4 \sqrt{3})$ be a point on the parabola $y^2=4 \mathrm{a} x$ 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 are...

Consider the circle \(C: x^2+y^2=4\) and the parabola \(P: y^2=8 x\). If the set of all values of \(\alpha\), for which three chords of the circle \(C\) on three distinct lines passing through the point \((\alpha, 0)\) are bisected by the p...

Let \(A, B\) and \(C\) be three points on the parabola \(y^2=6 x\) and let the line segment \(A B\) 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 perpend...

Let \(L_1, L_2\) be the lines passing through the point \(P(0,1)\) and touching the parabola \(9 x^2+12 x+18 y-14=0\). Let \(Q\) and \(R\) be the points on the lines \(L_1\) and \(L_2\) such that the \(\triangle P Q R\) is an isosceles tria...

Let a conic \(C\) pass through the point \((4,-2)\) and \(P(x, y), x \geq 3\), 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...

Let \(C\) be the circle of minimum area touching the parabola \(y=6-x^2\) and the lines \(y=\sqrt{3}|x|\). Then, which one of the following points lies on the circle \(C\) ?

Suppose \(\mathrm{AB}\) is a focal chord of the parabola \(y^2=12 x\) of length \(l\) and slope \(\mathrm{m}<\sqrt{3}\). If the distance of the chord \(\mathrm{AB}\) from the origin is \(\mathrm{d}\), then \(l \mathrm{~d}^2\) is equal to __...

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

Let the length of the focal chord PQ of the parabola \(y^2=12 x\) be 15 units. If the distance of \(\mathrm{PQ}\) from the origin is \(\mathrm{p}\), then \(10 \mathrm{p}^2\) is equal to __________.

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

Let \(P(\alpha, \beta)\) be a point on the parabola \(y^2=4 x\). If \(P\) also lies on the chord of the parabola \(x^2=8 y\) 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 $y^2=4 x$ from the centre of the circle $x^2+y^2-4 x-16 y+64=0$ is $\mathrm{d}$, then $\mathrm{d}^2$ is equal to :

Let the line $\mathrm{L}: \sqrt{2} x+y=\alpha$ pass through the point of the intersection $\mathrm{P}$ (in the first quadrant) of the circle $x^2+y^2=3$ and the parabola $x^2=2 y$. Let the line $\mathrm{L}$ touch two circles $\mathrm{C}_1$ ...

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

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

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 \(\mathrm{D}\). If the focus of the parabola \(y^{2}=4 \mathrm{ax}\) passing through \(\mathrm{D}\) is ...

Let the tangent to the curve \(x^{2}+2 x-4 y+9=0\) at the point \(\mathrm{P}(1,3)\) on it meet the \(y\)-axis at \(\mathrm{A}\). Let the line passing through \(\mathrm{P}\) and parallel to the line \(x-3 y=6\) meet the parabola $$y^{2}=4 x$...

Let \(\mathrm{y}=f(x)\) represent a parabola with focus \(\left(-\frac{1}{2}, 0\right)\) and directrix \(y=-\frac{1}{2}\). Then
\(S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}\) :

Let $\mathrm{S}$ be the set of all $\mathrm{a} \in \mathrm{N}$ such that the area of the triangle formed by the tangent at the point $\mathrm{P}(\mathrm{b}$, c), b, c $\in \mathbb{N}$, on the parabola $y^{2}=2 \mathrm{a} x$ and the lines $x...

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