Question

Find the equation for the ellipse that satisfies the given conditions $b =3$, $c =4$, center at the origin; foci on the $x-axis.$

Answer 1

It is given that $b= 3,$ $c = 4$, center at the origin; foci on the $x-axis.$
Since the foci are on the $x-axis$, the major axis is along the $x-axis.$
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, $b = 3$, $c= 4.$
It is known that

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

$∴ a^2 = 3^2 \+ 4^2$
$⇒a^2=25$
$⇒a = √25$
$⇒a= 5$

Thus, the equation of the ellipse is $\dfrac{x^2}{5^2} \+ \dfrac{y^2}{3^2}=1$ or $\dfrac{x^2}{25} +\dfrac{y^2}{9} = 1$