Chord of Contact to a Circle

Consider a circle S with centre at the origin and radius $4$. Four circles $A, B, C$ and $D$ each with radius unity and centres $\left(-3,0\right), \left(-1,0\right), \left(1,0\right)$ and $\left(3,0\right)$ respectively are drawn. $A$ chord $PQ$ of the circle $S$ touches the circle $B$ and passes through the centre of the circle $C$. If the length of this chord can be expresses as $\sqrt{x}$, find $x$.

Let $P$ be any point on the circle $x^2+y^2-2x-1=0$ and $C$ be its centre. Let $AB$ be the chord of contact of $P$ with respect to the circle $x^2+y^2-2x=0$. Then the locus of the circumcentre of the triangle $CAB$ is

The distance between the chords of contact of tangents to the circle $x^2+y^2+2gx+2fy+c=0$ from the origin and the point $(g, f)$ is :

If the chord of contact of tangents from a point $A$ to a given circle passes through $B$, then the circle with $AB$ as a diameter will

Let $AB$ be the chord of contact of the point $(5,-5)$ w.r.t the circle $x^2+y^2=5,$ then the locus of the orthocentre of $\Delta PAB$ where $P$ is any point moving on the circle is

If $\left(p,q\right)$ represents the point through which the chord of contact of pair of tangent for the circle $x^2+y^2=1$ always passes, when it is given that the pair of tangent is drawn from any point on the line $y=4-2x$, then the value of  $2p+4q$ is

If the tangents are drawn from any point on the line $x+y=3$ to the circle $x^2+y^2=9$, then the chord of contact always passes through a fixed point. Find that point

If the pair of tangents are drawn from  $O\left(0, 0\right)$ to the circle $x^2+y^2-6x-8y=-21$ meets the circle in $A$ and $B$, then length of $BA$ is

If tangent at $\left(1,2\right)$ to the circle $c_1:x^2+y^2=5$ intersects the circle $c_2:x^2+y^2=9$ at $A \& B$ and tangents at $A \& B$ to the second circle meet at point $C$, then the co-ordinates of $C$ is

The distance between the chords of contact of tangents to the circle, $x^2+y^2+2gx+2fy+c=0$ from the origin and the point $\left(g,f\right)$ is:

The chords of contact of the pair of tangents drawn from each point on the line $2x+y=4$ to circle $x^2+y^2=1$ pass through the fixed point:

Locus of the mid points of the chord of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, so that chord is always touching the circle $x^2+y^2=c^2$, $(c<a, c<b)$ is

The condition that the chord $x\cos \alpha +y\sin \alpha -p=0$ of $x^2+y^2-a^2=0$ may subtend a right angle at the centre of circle, is

A tangent at a point on the circle x² + y² = a² intersects a concentric circle C at two points P and Q. The tangents to the circle C at P and Q meet at a point on the circle x² + y² = b² then the equation of circle 'C' is :

If two chords of the circle $x^2+y^2-ax-by=0$, drawn from the point $\left(a,b\right)$ is divided by the $x-$axis in the ratio $2:1$ then :

The distance between the chords of contact of tangents to the circle $x^2+y^2+2gx+2fy+c=0$ from the origin and the point $(g, f)$ is :