The length of common chord of the circles ( x − a ) 2 + y 2 = a 2 and x 2 + ( y − b ) 2 = b 2 is Options: A . &nbsp 2 √ a 2 + b 2 B . &nbsp a b √ a 2 + b 2 C . &nbsp 2 a b √ a 2 + b 2 D . &nbsp 2 a b √ a 2 − b 2

The length of common chord of the circles (x−a)2+y2 =a2 and x2+(yȡ

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

Options:

A .

2 \sqrt{a^{2} + b^{2}}

B .

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

C .

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

D .

\frac{2 a b}{\sqrt{a^{2} - b^{2}}}

Answer: Option C
:
C

Equation of common chord is $a x - b y$ = 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 = $2 \sqrt{a^{2} - \frac{a^{4}}{a^{2} + b^{2}}}$ = $\frac{2 a b}{\sqrt{a^{2} + b^{2}}}$ .

Was this answer helpful ?

Next Question