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  4. The tangents at the extremities of the latus rectum of the ellipse 3x2+4y2=12 form a rhombus PQRS . Area (in sq. units) of the rhombus PQRS outside the ellipse is equal to

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Q. The tangents at the extremities of the latus rectum of the ellipse $3x^{2}+4y^{2}=12$ form a rhombus $PQRS$ . Area (in sq. units) of the rhombus $PQRS$ outside the ellipse is equal to

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Solution:

Solution
One of the end of latus rectum is $\left(1 , \frac{3}{2}\right)$
Tangent at $\left(1 , \frac{3}{2}\right)$ is $x+2y=4$
Area of rhombus $=4\left(\frac{1}{2} \cdot 4 \cdot 2\right)=16$
Hence, the required area $=16-2\sqrt{3}\pi $ sq. units