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Question

If the length of the tangent from $$(f, g)$$ to the circle $$x^{2} + y^{2} = 6$$ be twice the length of the tangent from $$(f, g)$$ to the circle $$x^{2} + y^{2} + 3x + 3y = 0$$, then will $$f^{2} + g^{2} + 4f + 4g + 2 = 0$$


Solution

We know that,
Length of Tangent from a Point (a,b) to a Circle $$x^2+y^2+2gx+2fy+c=0$$
is $$L=\sqrt{a^2+b^2+2ga+2fb+c}$$

So, According to Question
$$\sqrt{f^2+g^2-6}=2\sqrt{f^2+g^2+3f+3g}\Rightarrow f^2+g^2-6=4(f^2+g^2+3f+3g)\\\Rightarrow 3f^2+3g^2+12f+12g+6=0\\ \Rightarrow f^2+g^2+4f+4g+2=0$$

Maths

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