Question

Find the equation for the ellipse that satisfies the given conditions: Vertices $(0, ±13),$ foci $(0, ±5)$

Answer 1

Vertices $(0, ±13)$, foci $(0, ±5)$
Here, the vertices are on the $y-axis$.
Therefore, the equation of the ellipse will be of the form $\dfrac{x^2}{a^2} \+ \dfrac{y^2}{b^2} = 1$, where a is the semi-major axis.
Accordingly, $a = 13$ and $c = 5$.
It is known that

$a^2 = b^2 \+ c^2.$

$13^2 = b^2+5^2$

$169 = b^2 \+ 15$

$b^2 = 169 – 125$

$b = √144$

$= 12$

∴ The equation of the ellipse is $\dfrac{x^2}{12^2} \+ \dfrac{y^2}{13^2} = 1$ or $\dfrac{x^2}{144} + \dfrac{y^2}{169} = 1$ (Ans)