Interaction Between Two Curves

Let the parabolas $y=x^2+ax+b$ and $y=x\left(c-x\right)$ touch each other at $\left(1,0\right)$. Then

If $f\left(P\right)$ is the number of common tangents to two parabolas $x^2=2y$ and ${\left(y+\frac{1}{2}\right)}^2=4Px$

The equation of common tangent to both the curves $y^2=16x$ and ${\left(x+4\right)}^2+y^2=16$ is 

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

The circle $x^2+y^2=5$ meets the parabola $y^2=4x$ at $P$ and $Q$, then the length of $PQ$ ( in units ) is

The equation of the common tangent to the parabolas ${\mathrm{y}}^2=4\mathrm{ax}$ and ${\mathrm{x}}^2=4\mathrm{by}$ is given by

The equation of a common tangent to $y^{\mathit{2}}\mathit{=}\mathit{4}x$ and the curve $x^2+4y^2=8$ can be

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

Radius of largest circle which passes through the focus of the parabola $y^2=5x,$ and is contained in the parabola, is

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

The shortest distance between the curves $y=\left|x^2-6x-27\right|$ and ${\left(x-3\right)}^2+{\left(y-39\right)}^2=4$ is

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

Suppose $P$ and $Q$ be two distinct points on the parabola having equation $y^2=4x$ such that circle for which $PQ$ is diameter passes through vertex of the given parabola. If area of $OPQ$($O$ is origin) is $20$ sq. units and the diameter of circumcircle of $OPQ$ is $d$ units, then $\frac{d^2}{65}$=

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

If the minimum area of the circle which touches the parabola $y=x^2+1$ and $y^2=x-1$ is $\frac{a\pi }{b} \mathrm{sq}. \mathrm{units}$, where $a \& b$ are co-prime numbers, then the value of $a+b$ is

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