Parabola

If two tangents drawn from a point $P$ to the parabola ${y^2} = 4x$ are at right angles, then the locus of $P$ is

Let $O$ be the vertex and $Q$ be any point on the parabola, ${{x^2} = 8y}$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1:3$, then locus of $P$ is :

Tangents drawn from the point ($-$8, 0) to the parabola y² = 8x touch the parabola at $P$ and $Q.$ If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to :

If $a \ne 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas ${y^2} = 4ax$ and ${x^2} = 4ay$, then :

The length of the chord of the parabola x² $=$ 4y having equation x – $\sqrt 2 y + 4\sqrt 2 = 0$  is -

If one end of a focal chord of the parabola, y² = 16x is at (1, 4), then the length of this focal chord is :

Let the latus ractum of the parabola y² = 4x be the common chord to the circles C₁ and C₂ each of them having radius 2$\sqrt 5$. Then, the distance between the centres of the circles C₁ and C₂ is :

Let PQ be a focal chord of the parabola y² = 4x such that it subtends an angle of ${\pi \over 2}$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$. If e is the eccentricity of the ellipse E, then the value of ${1 \over {{e^2}}}$ is equal to :

Let the focal chord of the parabola $\mathrm{P}: y^{2}=4 x$ along the line $\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $\mathrm{H}: x^{2}-y^{2}=4$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is :

Let $\mathrm{PQ}$ be a focal chord of the parabola $y^{2}=36 x$ of length 100 , making an acute angle with the positive $x$-axis. Let the ordinate of $\mathrm{P}$ be positive and $\mathrm{M}$ be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $\mathrm{PQ}$?

Let L be a tangent line to the parabola y² = 4x $-$ 20 at (6, 2). If L is also a tangent to the ellipse ${{{x^2}} \over 2} + {{{y^2}} \over b} = 1$, then the value of b is equal to :

If $y = {m_1}x + {c_1}$ and $y = {m_2}x + {c_2}$, ${m_1} \ne {m_2}$ are two common tangents of circle ${x^2} + {y^2} = 2$ and parabola y² = x, then the value of $8|{m_1}{m_2}|$ is equal to :

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 :

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 $\mathrm{Q}$ ?

Two common tangents to the circle ${x^2} + {y^2} = 2{a^2}$ and parabola ${y^2} = 8ax$ are :

Given : A circle, $2{x^2} + 2{y^2} = 5$ and a parabola, ${y^2} = 4\sqrt 5 x$. Statement-1 : An equation of a common tangent to these curves is $y = x + \sqrt 5$. Statement-2 : If the line, $y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right)$ is their common tangent, then $m$ satiesfies ${m^4} - 3{m^2} + 2 = 0$.

If the common tangents to the parabola, x² = 4y and the circle, x² + y² = 4 intersect at the point P, then the distance of P from the origin, is :

Two parabolas with a common vertex and with axes along x-axis and $y$-axis, respectively intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $3$, then the equation of the common tangent to the two parabolas is :

Equation of a common tangent to the circle, x² + y² – 6x = 0 and the parabola, y² = 4x is :