Question

The length of common chord of the circles $(x-a{)}^2+y^2$ =$a^2$ and $x^2+(y-b{)}^2$ = $b^2$ is

• A. $2\sqrt{a^2+b^2}$
• B. $\frac{ab}{\sqrt{a^2+b^2}}$
• C. $\frac{2ab}{\sqrt{a^2+b^2}}$
• D. $\frac{2ab}{\sqrt{a^2-b^2}}$

Answer 1

The correct option is C

$\frac{2ab}{\sqrt{a^2+b^2}}$

Equation of common chord is $ax-by$ = 0

Now length of common chord

= $2\sqrt{r_1^2-p_1^2}$ = $2\sqrt{r_2^2-p_2^2}$

Where $r_1$ and $r_2$ are radii of given circles and $p_1,p_2$ are the perpendicular distances from centres of circles to common chords.

Hence required length = $2\sqrt{a^2-\frac{a^4}{a^2+b^2}}$ = $\frac{2ab}{\sqrt{a^2+b^2}}$.