Find the equation of the ellipse, whose length of the major axis is $20$ and foci are $(0,\,\pm 5)$
Find the equation of the ellipse, whose length of the major axis is $20$ and foci are $(0,\,\pm 5)$
since the foci are on $y-$ axis, the major axis is along the $y-$ axis. So, equation of the cllipse is of the form $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$
Given that
$a=$ semi-major axis $=\frac{20}{2}=10$
and the relation $c^{2}=a^{2}-b^{2}$ gives
$5^{2}=10^{2}-b^{2} $ i.e., $b^{2}=75$
Therefore, the equation of the ellipse is
$\frac{x^{2}}{75}+\frac{y^{2}}{100}=1$
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