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Director Circle

If the length...

Question

If the length of the tangent from (h, k) to the circle $x^{2} + y^{2} = 16$ is twice the length of the tangent from the same point to the circle $x^{2} + y^{2} + 2 x + 2 y = 0$ , then

h^{2} + k^{2} + 4 h + 4 k + 16 = 0

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h^{2} + k^{2} + 3 h + 3 k = 0

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3 h^{2} + 3 k^{2} + 8 h + 8 k + 16 = 0

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3 h^{2} + 3 k^{2} + 4 h + 4 k + 16 = 0

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Solution

The correct option is B $3 h^{2} + 3 k^{2} + 8 h + 8 k + 16 = 0$
Length of tangent = $\sqrt{s_{1}}$
$s = x^{2} + y^{2} - 16$
$s_{1} = h^{2} + k^{2} - 16$ ..(1)
for circle $x^{2} + y^{2} + 2 x + 2 y = 0$
$s_{1} = h^{2} + k^{2} + 2 h + 2 k$ ...(2)
It given that
$\sqrt{h^{2} + k^{2} - 16} = 2 \sqrt{h^{2} + k^{2} + 2 h + 2 k}$
$\to h^{2} + k^{2} - 16 = 4 h^{2} + 4 k^{2} + 8 h + 8 k$
$\to 3 h^{2} + 3 k^{2} + 8 h + 8 k + 16 = 0$

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