Question

Find the equation for the ellipse that satisfies the given conditions: Major axis on the $x$-axis, centre is origin and passes through the points $(4,3)$ and $(6,2).$

Answer 1

Since the major axis is on the $x$-axis, the equation of the ellipse will be of the form,
$\frac{x^2}{a}+\frac{y^2}{b^2}=1...\left(1\right)$
Where $a$ is the semi-major axis.
The ellipse passes through points $(4,3)$ and $(6,2)$. Hence,
$\frac{16}{a^2}+\frac{9}{b^2}=1...\left(2\right)$
$\frac{36}{a^2}+\frac{4}{b^2}=1...\left(3\right)$
On solving equations (2) and (3) we obtain $a^2=52\text{and}{b}^2=13$
Thus the equation of the ellipse is $\frac{x^2}{52}+\frac{y^2}{13}=1$ or $x^2+4y^2=52$