Question:
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
$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$
Where, $a$ is the semi-major axis
The ellipse passes through points (3, 2) and (1, 6). Hence,
$\frac{9}{b^{2}}+\frac{4}{a^{2}}=1$
$\frac{1}{b^{2}}+\frac{36}{a^{2}}=1$
On solving equations $(2)$ and $(3)$, we obtain $b^{2}=10$ and $a^{2}=40$.
Thus, the equation of the ellipse is $\frac{x^{2}}{10}+\frac{y^{2}}{40}=1$ or $4 x^{2}+y^{2}=40$.