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Orthogonality of Two Circles

Find the valu...

Question

Find the value of k, if the circles $x^{2} + y^{2} + 4 x + 8 = 0$ and $x^{2} + y^{2} - 16 x + k = 0$ are orthogonal.

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Solution

Fact:
Two circles are orthogonal if $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

Here $g_{1} = 2, f_{1} = 0, c_{1} = 8, g_{2} = - 8, f_{2} = 0, c_{2} = k$
$∴ 2 (2) (- 8) + 2 (0) (0) = 8 + k$
$\Rightarrow - 32 = k + 8$
$\Rightarrow k = - 32 - 8 = - 40$

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