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Length of Common Chord

l and m are...

Question

$l$ and $m$ are the lengths of the tangents from the origin and the point $(9, - 1)$ to the circle $x^{2} + y^{2} + 18 x - 2 y + 32 = 0$ . The value of $17 (l^{2} + m^{2})$ is equal to

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Solution

Given equation of circle is
$x^{2} + y^{2} + 18 x - 2 y + 32 = 0$
Square of length of tangent from (0,0) is
$l^{2} = 32$
Square of length of tangent from (9,-1) is
$m^{2} = 81 + 1 + 162 + 2 + 32 = 278$
So, $17 (l^{2} + m^{2}) = 17 (310) = 5270$

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