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____.

The equation of circle such that the length of the tangent to it from the points $(1,0),\left(2,0\right)$ and $\left(3,2\right)$ are $1,$ $\sqrt{7}\text{,} \sqrt{2}$ respectively, is

Let $2{\text{x}}^2+{\text{y}}^2-3\text{xy}=0$ be the equation of a pair of tangents drawn from the origin 'O' to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

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

Let $x^2+y^2-4x-2y-11=0$ be a circle. A pair of tangents from the point $(4, 5)$ with a pair of radii form a quadrilateral of the area:

Find the inverse of the point $\left(1,2\mathrm{ }\right)$ with respect to the circle $x^2+y^2-4x-6y+9=0$ 

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 $x^2+y^2=4$ 

The lengths of the tangents from the point$(1, 2)$ to the circle $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-k=0$ are in the ratio $4 : 3,$ then the value of $k$ is

If the lengths of the tangents drawn from the point $(1, 2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-\lambda =0$ are in the ratio $3:4,$ then $\lambda =$ _____

The angle between the two tangents drawn from origin to the circle $x^2+y^2-14x+2y+25=0$ is $\_\_\_\_\_\_\_\_\_$

The locus of the mid points of the chords of the circle $x^2+y^2-2bx=0$ passing through the origin is

Let $T$ be a circle with diameter $AB$ and centre $O$. Let $l$ be the tangent to $T$ at $B$. For each point $M$ on $T$ different from $A$, consider the tangent $t$ at $M$ and let interest $l$ at $P.$ Draw a line parallel to $AB$ through $P$ intersecting $OM$ at $Q.$ The locus of $Q$ as $M$ varies over $T$ is

If $P\left(x_1,y_1\right)$ is a point such that the lengths of the tangents from it to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ are in the ratio $2:3$, then the locus of $P$ is

If the lengths of the tangents drawn from $P$ to the circles $x^2+y^2-2x+4y-20=0$ and $x^2+y^2-2x-8y+1=0$ are in the ratio $2:1$, then the locus of $P$ 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