Chapter 12, Problem 7CR

To determine

The equation of an ellipse for the given information as center at point $\left(0,0\right)$, focus at point $\left(0,3\right)$ and a vertex at point $\left(0,4\right)$.

Answer to Problem 7CR

Solution:

The equation of an ellipse for which center is at origin, focus at point $\left(0,3\right)$ and a vertex at point $\left(0,4\right)$ is $\frac{x^2}{7}+\frac{y^2}{16}=1$.

Explanation of Solution

Given information:

Center of the circle is at $\left(0,0\right)$, focus at point $\left(0,3\right)$ and a vertex at point $\left(0,4\right)$.

Explanation:

The $x$-co-ordinates center, focus and vertex is $0$. Thus major axis is the $y$-axis.

Therefore, $a=4,c=3$.

To find $b$ using formula $b^2=a^2-c^2$,

$b^2=4^2-3^2=16-9=7$

$b^2=7\Rightarrow b=\sqrt{7}$

Thus using values of $a=4$ and $b=\sqrt{7}$ the equation of ellipse is $\frac{x^2}{7}+\frac{y^2}{16}=1$.