Chapter 12, Problem 7CR

To determine

To find: The equation of an ellipse.

Answer to Problem 7CR

The equation of the ellipse is $\frac{x^2}{7}+\frac{y^2}{16}=1$ .

Explanation of Solution

Given information:

The given focus at $\left(0,3\right)$ and a vertex at $\left(0,4\right)$

Calculation:

The distance from the centre to a focus of the ellipse $\left(0,3\right)$ is $c=3$ . The distance of the centre a vertex of the ellipse $\left(0,4\right)$ is $a=4$

The general equation of the ellipse when the major axis is the y-axis is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ when a $>b>0$ and $b^2=a^2-c^2$

Substitute values are $4$ for $a$ and $3$ for $c$ in this equation to get $b^2$ .

  $\begin{array}{c}{b}^2=4^2-3^2\\ =7\end{array}$

The sequence root of the both sides,

  $\begin{array}{c}\sqrt{b^2}=\sqrt{7}\\ b=\pm \sqrt{7}\end{array}$

Substitute the values of $4$ for $a$ and $3$ for $c$ in the equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ .

  $\frac{x^2}{7}+\frac{y^2}{16}=1$ .

Therefore, the equation of the ellipse is $\frac{x^2}{7}+\frac{y^2}{16}=1$ .