Question

The equation of circle with centre $(-a,-b)$ and radius $\sqrt{a^2-b^2}$is

• A. $x^2+y^2+2ax+2by+2b^2=0$
• B. $x^2+y^2–2ax–2by–2b^2=0$
• C. $x^2+y^2–2ax–2by+2b^2=0$
• D. $x^2+y^2–2ax+2by+2a^2=0$

Answer 1

The correct option is A

$x^2+y^2+2ax+2by+2b^2=0$

Explanation for the correct option:

Finding the equation of the circle:

We know that the equation of circle with centre $(x_1,y_1)$and radius $r$ is

${\left(x-x_1\right)}^2+{\left(y-y_1\right)}^2=r^2\dots \dots \left(1\right)$

According to the given data,

$x_1=-a,y_1=-b\text{and}r=\sqrt{a^2-b^2}$

Now substituting the values in equation $\left(1\right)$ we get,

$$\begin{gathered} \Rightarrow {\left[x-\left(-a\right)\right]}^2+{\left[y-\left(-b\right)\right]}^2={\left(\sqrt{a^2-b^2}\right)}^2 \\ \Rightarrow x^2+a^2+2ax+y^2+b^2+2by=a^2-b^2 \\ \Rightarrow \begin{array}{l}{x}^2+y^2+2ax+2yb+2b^2=0\end{array} \end{gathered}$$

Hence, option (A) is the correct answer.