Find length of intercept cut off by well $x^{2} + y^{2} + 4 x - 2 y - 3 = 0$ from line $x - y + 2 = 0$ .

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Solution

$x^{2} + y^{2} + 4 x - 2 y - 3 = 0$
$x = y - 2$
So, $(y - 2)^{2} + y^{2} + 4 (y - 2) - 2 y - 3 = 0$
$⟹ 2 y^{2} - 2 y - 7 = 0$
Let the root be $y_{1}, y_{2}$
So, $| y_{1} - y_{2} | = \sqrt{(y_{1} + y_{2})^{2} - 4 y_{1} y_{2}} = \sqrt{(\frac{- 2}{3})^{2} - 4 (\frac{- 7}{2})}$
$| y_{1} - y_{2} | = \sqrt{15}$
Length of the intercept $= \sqrt{| x_{1} - x_{2} |^{2} + | y_{1} - y_{2} |^{2}}$
$= \sqrt{(\sqrt{15})^{2} + (\sqrt{15})^{2}} = 15 \sqrt{2}$

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