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Find the equation for the ellipse that satisfies the given conditions: Centre at $(0,\,0),$ major axis on the $y-$ axis and passes through the points $(3,\,2)$ and $(1,\,6)$
Solution
since the centre is at $(0,\,0)$ and the major axis is on the $y-$ axis, the equation of the ellipse will be of the form
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ……….. $(1)$
Where, a is the semi-major axis The ellipse passes through points $(3,\,2)$ and $(1,\,6) .$ Hence,
$\frac{9}{b^{2}}+\frac{4}{a^{2}}=1$ ……….. $(2)$
$\frac{1}{b^{2}}+\frac{36}{a^{2}}=1$ ……….. $(3)$
On solving equations $(2)$ and $(3),$ we obtain $b^{2}=10$ and $a^{2}=40$.
Thus, the equation of the ellipse is $\frac{x^{2}}{10^{2}}+\frac{y^{2}}{40}=1$ or $4 x^{2}+y^{2}=40$
