Find the equation and length of the common chord of the two circles:
$x^{2} + y^{2} + 3 x + 5 y + 4 = 0$ and
$x^{2} + y^{2} + 5 x + 3 y + 4 = 0$ .

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Solution

To obtain equation of common chord of two circles, just subtract both equations

$\Rightarrow$ Equation of common chord= $- 2 x + 2 y + 0 = 0$

$\Rightarrow$ Equation of common chord = $x = y$

Lenght of common chord= $\frac{2 r_{1} r_{2}}{d}$ ; d is distance between centers

Here $r_{1} = r_{2} = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}}$

$d = \sqrt{2}$

$\Rightarrow$ Lenght of common chord = $\frac{2 \times \frac{25}{2}}{\sqrt{2}} = \frac{25}{\sqrt{2}}$

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