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Planet $M$ orbits around its sun, $S$, in an elliptical orbit with the sun at one of the foci. When $M$ is closest to $S$, it is $2\,unit$ away. When $M$ is farthest from $S$, it is $18\, unit$ away, then the equation of motion of planet $M$ around its sun $S$, assuming $S$ at the centre of the coordinate plane and the other focus lie on negative $y-$ axis, is
$\frac{{{x^2}}}{{36}} + \frac{{{{\left( {y - 8} \right)}^2}}}{{100}} = 1$
$\frac{{{x^2}}}{{36}} + \frac{{{{\left( {y + 8} \right)}^2}}}{{100}} = 1$
$\frac{{{x^2}}}{{64}} + \frac{{{{\left( {y - 8} \right)}^2}}}{{100}} = 1$
$\frac{{{x^2}}}{{64}} + \frac{{{{\left( {y + 8} \right)}^2}}}{{100}} = 1$
Solution
$b=10 $ and $ b e=8$
$\Rightarrow \quad e=\frac{4}{5}$ and $a=6$
Let ellipse is $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{(\mathrm{y}+8)^{2}}{\mathrm{b}^{2}}=100$
where $b=10 $ and $ a=6$
