The lengths of the tangents from any point of the circle $15 x^{2} + 15 y^{2} - 48 x + 64 y = 0$ to the two circle $5 x^{2} + 5 y^{2} - 24 x + 32 y + 75 = 0, 5 x^{2} + 5 y^{2} - 48 x + 64 y + 300 = 0$ are in the ratio?

1 : 2

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2 : 3

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3 : 4

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None of these

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Solution

The correct option is A $1 : 2$
The circle $15 x^{2} + 15 y^{2} - 48 x + 64 y = 0$ passes through $(0, 0)$ . Hence, let the point be origin from which the tangents are drawn to circle.

Length of tangent to circle: $5 x^{2} + 5 y^{2} - 24 x + 32 y + 75 = 0$
$L_{1} = \sqrt{S_{11}} = \sqrt{75} = 5 \sqrt{3}$
Length of tangent to circle: $5 x^{2} + 5 y^{2} - 48 x + 64 y + 300 = 0$
$L_{2} = \sqrt{S_{11}} = \sqrt{0 + 0 + 300} = 10 \sqrt{3}$
Hence, $\frac{L_{1}}{L_{2}} = \frac{5 \sqrt{3}}{10 \sqrt{3}} = \frac{1}{2}$

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