Equation of the circle whose radius is $a+b$ and centre (a,-b)

Fact:

Equation of circle with centre $(\mathrm{h}, \mathrm{k})$ and having radius $r$ is given by,

$(x-h)^{2}+(y-k)^{2}=r^{2}$

Here, centre is $(\mathrm{a},-\mathrm{b}$ ) and radius is $\mathrm{a}+\mathrm{b}$ hence equation of required circle is given by,

$\begin{array}{l} (x-a)^{2}+[y-(-b)]^{2}=(a+b)^{2} \\ \Rightarrow(x-a)^{2}+(y+b)^{2}=a^{2}+b^{2}+2 a b \\ \Rightarrow x^{2}-2 a x+a^{2}+y^{2}+2 b y+b^{2}=a^{2}+b^{2}+2 a b \\ \Rightarrow x^{2}+y^{2}-2 a x+2 b y=2 a b \end{array}$