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Q.
The area of the quadrilateral with its vertices at the foci of the conics $9 x^{2}-16 y^{2}-18 x+32 y-23=0$ and $25 x^{2}+9 y^{2}-50 x-18 y+33=0$, is
Application of Integrals
Solution:
$1^{\text {st }}$ is a hyperbola
$9(x-1)^{2}-16(y-1)^{2}=16$ with $e=5 / 4$
and $2^{\text {nd }}$ is an ellipse
$25(x-1)^{2}+9(y-1)^{2}=1$ with $e=4 / 5$
with $x-1=X$ and $y-1=Y$
area $=\frac{1}{2} d_{1} d_{2}=\frac{1}{2} \cdot \frac{10}{3} \cdot \frac{8}{15}=\frac{8}{9}$
$e_{E} \cdot e_{H}=1$
