Find the length of the chord intercepted by the circle $x^{2} + y^{2} - x + 3 y - 22 = 0$ on the line $y = x - 3$ .

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Solution

To find the co-ordinates of the end of the chords, let us substitute the equation of the line in the circle.
Hence
$x^{2} + (x - 3)^{2} - x + 3 (x - 3) - 22 = 0$
$\Rightarrow x^{2} + x^{2} - 6 x + 9 - x + 3 x - 9 - 22 = 0$
$\Rightarrow 2 x^{2} - 4 x - 22 = 0$
$\Rightarrow x^{2} - 2 x - 11 = 0$
$\Rightarrow x = \frac{2 \pm \sqrt{4 + 44}}{2}$
Hence $x = 1 \pm 2 \sqrt{3}$
Therefore $x_{1} = 1 + 2 \sqrt{3}$ and $x_{2} = 1 - 2 \sqrt{3}$ .
Hence $y_{1} = - 2 + 2 \sqrt{3}$ and $y_{2} = - 2 - 2 \sqrt{3}$
Hence the length of the chord is
$D = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
$= \sqrt{(4 \sqrt{3})^{2} + (4 \sqrt{3})^{2}}$
$= \sqrt{96}$
$= 4 \sqrt{6}$ units.

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