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  4. The common tangent to the circles x2 + y2 = 4 and x2 + y2 + 6x + 8y - 24 = 0 also passes through the point :

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

JEE MainJEE Main 2019Conic Sections

Solution:

Circle touches internally
$C_1(0, 0); r_1 = 2$
$C_2 : (-3, -4); r_2 = 7$
$C_1C_2 = |r_1 - r_2|$
$S_1 - S_2 = 0 \; \Rightarrow $ eqn. of common tangent
$6x + 8y - 20 = 0$
$3x + 4y = 10$
$(6, -2)$ satisfy it