Find the length of the chord $x - y - 3 = 0$ of the circle $x^{2} + y^{2} - x + 3 y - 22 = 0$

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Solution

Center of circle $= (\frac{1}{2}, \frac{- 3}{2}$ )
R $=$ Radius of circle $= \sqrt{\frac{1}{4} + \frac{9}{4} + 22}$
D $=$ Distance of center from chord $= \frac{| \frac{1}{2} + \frac{3}{2} - 3 |}{\sqrt{2}}$ $= \frac{1}{\sqrt{2}}$
L $= 2 \times$ $\sqrt{R^{2} - D^{2}}$
L $=$ Length of chord $= 4 \sqrt{6}$

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