Interaction between Two Conics

A circle described on the latus rectum of the parabola $y^2=4x$ as a diameter meets the axis at

The equation of a common tangent to the curves $y^2=4x$ and $x^2+2y^2=2$ is

Let $P$ and $Q$ be distinct points on the parabola ${\mathrm{y}}^2=2\mathrm{x}\mathrm{ }$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\mathrm{\Delta }OPQ$ is $3\sqrt{2},\mathrm{ }$ then the coordinates of $P$ can be

The equation of the common tangent to the curves $y^2=4x$ and $x^2+32y=0$ is $x+by+c=0$. The value of $\left|{\sin }^{–1}\left(\sin 1\right)+{\sin }^{–1}\left(\sin b\right)+{\sin }^{–1}\left(\sin c\right)\right|$ is equal to

A circle passes through the points of intersection of the parabola $\mathrm{y}+1=(\mathrm{x}-4{)}^2$ and $\mathrm{x}$-axis. Then the length of tangent from origin to the circle is

Two distinct parabolas have the same focus and co-ordinate axes as their directrices respectively, then slope of their common chords are

If a circle is given by the equation $x^2+y^2+2\lambda x=0, \lambda \in R,$ which touches the parabola $y^2=4x$ externally. Then, 

Let A be a circle ${\mathrm{x}}^2+{\mathrm{y}}^2=16$ and B be a parabola ${\mathrm{y}}^2=4\mathrm{x}$ . Then the number of points with integral co-ordinates that lie in the interior of the region common to both A and B is

If the shortest distance between $y^2=4x$ and  $x^2+y^2-4x-16y+63=0$ is $\sqrt{p}$ units, then the value of $p$ is

Which of the following is compound?

If two parabolas $y^2=4a\left(x–k\right)$ and $x^2=4a\left(y–k\right)$ have only one common point $P$, then the equation of normal to $y^2=4a\left(x–k\right)$ at $P$ is

If the line $y-2=0$ is the directrix of the parabola $x^2-ky+32=0, k\ne 0$ and the parabola intersects the circle $x^2+y^2=8$ at two real distinct points, then the absolute value of $k$ is

Maximum number of points on parabola $y^2=16x$ which are equidistant from a variable point $P$ (which lie inside the parabola), is/are:

The shortest distance between the circle $x^2+y^2=8$ and the curve $2\left[{\left(x-3\right)}^2+{\left(y+3\right)}^2\right]={\left(x-y-2\right)}^2$ is

The common tangent of the parabolas $y^2=4x$ and $x^2=-8y$ is

The points of intersection of the parabolas $y^2=5x$ and $x^2=5y$ lie on the line

A circle of radius 4, drawn on a chord of the parabola $y^2=8x$ as diameter, touches the axis of the parabola. Then, the slope of the chord is

The coordinates of the point on the parabola ${\mathrm{y}}^2=8\mathrm{x}$ , which is at minimum distance from the circle ${\mathrm{x}}^2\mathrm{ }+\mathrm{ }{\left(\mathrm{y}\mathrm{ }+\mathrm{ }6\right)}^2=1$ are-

The shortest distance between the parabolas $y^2=4x$ and $y^2=2x-6$ is

The shortest distance between the parabolas $y^2=4x$ and $y^2=2x-6$ is