Pair of Tangents, Chord of Contact and Chord with Midpoint of a Circle

The angle between the tangents drawn from origin to the circle ${\left(x-7\right)}^2+{\left(y+1\right)}^2=25$ is equal to____.

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 $x$.

The area of the triangle formed by the tangents from the point $(4, 3)$ to the circle $x^2+y^2=9$ and the line joining their points of contact is

The polars of two points $A(5,7)$ and $B(3,3)$ with respect to the circle $x^2+y^2+2gx+2fy+c=0$ intersect at $C$ then find the polar of $C$ with respect to the circle.

The polars of two points $A(1,3)$ and $B(2,1)$ with respect to the circle $x^2+y^2+2gx+2fy+c=0$ intersect at $C$ then find the polar of $C$ with respect to the circle.

Tangents are drawn from the point $P\left(1,-1\right)$ to the circle $x^2+y^2-4x-6y-3=0$ with centre $C$. $A$ and $B$ are points of contact

The equation of the circle circumscribing the triangle formed by pair of tangent and corresponding chord of contact, is

The area of triangle formed by pair of tangents drawn from $\left(1,-1\right)$ to the circle $x^2+y^2-4x-6y-3=0$  and corresponding chord of contact, is 

Find the length of chord of contact drawn to the circle $x^2+y^2=4$ and the perpendicular distance of the chord from the origin is $\frac{10}{\sqrt{45}}$

Find the length of chord of contact drawn to the circle $x^2+y^2=10$ and the perpendicular distance of the chord from the origin is $\frac{10}{\sqrt{45}}$

Identify the inverse point of $(-2,-2)$ with respect to the circle $x^2+y^2=4$ 

Identify the inverse point of $(3,-3)$ with respect to the circle $x^2+y^2=9$ 

Find the inverse point of $(-3,-3)$ with respect to the circle $x^2+y^2=9$ 

Find the inverse point of $(2,-2)$ with respect to the circle $x^2+y^2=4$ 

If $A$ and $B$ are the point of contact of pair of tangents drawn from $P\left(6,8\right)$ on the circle $x^2+y^2=16$, then the circumradius of $\Delta OAB$ (where $O$ being origin) is

Tangents drawn from the point $\mathrm{P}\left(1, 8\right)$ to the circle $x^2+y^2-6x-4y-11=0$ touch the circle at the points $\mathrm{A}$ and $\mathrm{B}.$ The equation of the circumcircle of the triangle $\mathrm{PAB}$ is

Tangents drawn from the point $P\left(1,8\right)$ to the circle $x^2+y^2-6x-4y-11=0$ touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $PAB$ is

The angle between a pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9{\sin }^2\alpha +13{\cos }^2\alpha =0$ is $2\alpha$.
The equation of the locus of point P is $x^2+y^2+ax+by+c=0.$ Evaluate $b^2-ac$.

A regular hexagon is formed by two equilateral triangles inscribed in the circle $x^2+y^2=4$. If $S$ is the area of the hexagon (in sq. units), then find the greatest integer contained in $S$.

The polars of a point $\mathrm{P}$ with respect to two fixed circles meet in the point $\mathrm{Q}$. Prove that the circle on $\mathrm{P}\mathrm{Q}$ as diameter passes through two fixed points, and cuts both the given circles at right angles.

The locus of the point of intersection of tangents to the circle x = a cosθ, y = a sinθ at a point whose parametric angles differ by π/3 is x² + y² = ka². Find the value of 3k.