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

It is given that $b=3,ae=4$, centre at the origin; foci on the $x$- axis.
Since the foci are on the $x$-axis $\Rightarrow$ the major axis is along the $x$-axis.
Therefore, the equation of the ellipse will be of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$  where $a$ is the semi-major axis.
Accordingly $b=3,ae=4.$
It is known that $a^2=b^2+a{e}^2$
$\therefore a^2=3^2+4^2=9+16=25$
$\Rightarrow a=5$
Thus, the equation of the ellipse is $\frac{x^2}{5^2}+\frac{y^2}{3^2}=1$
$\Rightarrow \frac{x^2}{25}+\frac{y^2}{9}=1.$