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).

Answer 1

Given that, since the center is at $(0, 0)$ and the major axis is on the $y-axis$, the equation of the ellipse will be of the form

$\dfrac{x^2}{b^2} \+ \dfrac{y^2}{a^2} = 1 ....(1)$, (where ‘a’ is the semi-major axis).
The ellipse passes through points $(3, 2)$ and $(1, 6)$. Hence,
$\dfrac{9}{b^2} \+ \dfrac{4}{a^2}…. (2)$

\dfrac{1}{b^2} \+ \dfrac{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 $\dfrac{x^2}{10} \+ \dfrac{y^2}{40} = 1$ or $4x^2 \+ y^2 = 40$