Question

Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0,±6)

Answer 1

Length of minor axis $= 16$ ; foci $= (0, ±6)$.
Since the foci are on the y-axis, the major axis is along 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, $2b = 16$
$⇒ b = 8$ and $c = 6.$

It is known that

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

$a2 = 8^2 \+ 6^2$

$= 64 + 36$

$=100$

$a = √100$

$a = 10$

Thus, the equation of the ellipse is $\dfrac{x^2}{8^2} \+ \dfrac{y^2}{10^2} = 1$or $\dfrac{x^2}{64} + \dfrac{y^2}{100} = 1$ (Ans.)