The value of $k$ for which two tangents can be drawn from $(k, k)$ to the circle $x^{2} + y^{2} + 2 x + 2 y 16 = 0$ is

k ϵ R^{+}

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k ϵ R

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k ϵ (- \infty, - 4) \cup (2, \infty)

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k ϵ (0, 1]

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Solution

The correct option is C $k ϵ (- \infty, - 4) \cup (2, \infty)$
For two tangents to be drawn to $C (x, y) :$

$⟹ x^{2} + y^{2} + 2 x + 2 y - 16 = 0$

From P(k,k) , the point P must lie outside the circle

$⟹ C (k, k) > 0$

$⟹ k^{2} + k^{2} + 2 k + 2 k - 16 > 0$

$⟹ k^{2} + 2 k - 8 > 0$

$⟹ (k + 4) (k - 2) > 0$

$⟹ k \in (- \infty, - 4) \cup (2, \infty)$

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