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Number of Common Tangents to Two Circles in Different Conditions

The common ta...

Question

The common tangent to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} + 6 x + 8 y - 24 = 0$ also passes through the point:

A

(- 4, 6)

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B

(6, - 2)

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C

(- 6, 4)

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D

(4, - 2)

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Solution

The correct option is B $(6, - 2)$
Circle touches internally
$C_{1} (0, 0) : r_{1} = 2$
$C_{2} : (- 3, - 4); r_{2} = 7$
$C_{1} C_{2} = | r_{1} - r_{2} |$
$S_{1} - S_{2} = 0$ $\Rightarrow$ equation of common tangent
$6 x + 8 y - 20 = 0$
$3 x + 4 y = 10$
$(6, - 2)$ satisfies the equation.

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