# Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0),

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 $4 x^{2}+y^{2}=40$.