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  4. A rhombus is inscribed in the region common to the two circles x2+y2-4 x-12=0 and x2+y2+4 x-12=0 with two of its vertices on the line joining the centers of the circles. The area of rhombus is

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Q. A rhombus is inscribed in the region common to the two circles $x^{2}+y^{2}-4 x-12=0$ and $x^{2}+y^{2}+4 x-12=0$ with two of its vertices on the line joining the centers of the circles. The area of rhombus is

ManipalManipal 2014

Solution:

Circles with centers $A(-2,0)$ and $B(+2,0)$, each of radius $4$ has $Y$-axis as their common chord.
image
$\Rightarrow \Delta A B C$ is an equilateral triangle.
Hence, the area of rhombus $A D B C$ is
$=2 \cdot \frac{\sqrt{3}}{4} \times 4^{2}=8 \sqrt{3}$