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

Given: Conditions for ellipse: $b=3,c=4$, center at the origin: foci on the $x$-axis.
$\because$ Foci are on $x$-axis.
$\therefore$ The major axis is along the $x$-axis.
Hence, equation of ellipse is of the form: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
We know, $c^2=a^2-b^2$
$\Rightarrow 4^2=a^2-3^2$
$\Rightarrow 16=a^2-9$
$\Rightarrow a^2=16+9$
$\Rightarrow a^2=25$
$a=5,b=3$
Putting value of $a$ and $b$ in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$\therefore$ Required equation of the ellipse is $\frac{x^2}{5^2}+\frac{y^2}{3^2}=1$ or $\frac{x^2}{25}+\frac{y^2}{9}=1$