G Tewani Solutions for Chapter: Circles, Exercise 4: DPP

Two circles touch the $x$-axis and the line $y=mx$. They meet at $\left(9,6\right)$ and at one more point. If the product of their radii is $\frac{117}{2}$, then the value of $m$ is

Tangents drawn from $P\left(1, 8\right)$ to the circle $x^2+y^2-6x-4y-11=0$ touches the circle at the points $A$ and $B$, respectively. The radius of the circle which passes through the points of intersection of circles $x^2+y^2-2x-6y+6=0$ and $x^2+y^2+2x-6y+6=0$ and intersects the circumcircle of the $∆PAB$ orthogonally is equal to

If the radius of the circle touching the pair of lines $7x^2-18xy+7y^2=0$ and the circle $x^2+y^2-8x-8y=0$ and contained in the given circle is equal to $k$, then $k^2$ is equal to

Equation of a circle having radius equal to twice the radius of the circle $x^2+y^2+\left(2p+3\right)x+\left(3-2p\right)y+p-3=0$ and touching it at the origin is

The real numbers $a$ and $b$ are distinct. Consider the circles ${\omega }_1: {\left(x-a\right)}^2+{\left(y-b\right)}^2=a^2+b^2$ and ${\omega }_2: {\left(x-b\right)}^2+{\left(y-a\right)}^2=a^2+b^2.$ Which of the following is/are true?

Consider two circles $S_1\equiv x^2+y^2+8x=0$ and $S_2\equiv x^2+y^2-2x=0$. Let $∆PQR$ be formed by the common tangents to circles $S_1$ and $S_2$. Then which of the following hold(s) good?
 

A circle touches the line $x+y-2=0$ at $\left(1, 1\right)$ and cuts the circle $x^2+y^2+4x+5y-6=0$ at $P$ and $Q$. Then:

A circle $S=0$ passes through the common points of family of circles $x^2+y^2+\lambda x-4y+3=0$, and $\left(\lambda \in R\right)$ has minimum area, then