Question

Finding the Equation of an Ellipse

Find an equation for the ellipse that satisfies the given conditions.

Foci: $(0,\pm 2)$, length of minor axis: 6

Short answer

The equation of ellipse is $\frac{x^2}{9}+\frac{y^2}{13}=1$

Step-by-step solution

Step 1. Given information

Foci $(0,\pm 2)$ The length of the minor axis is 6.

Step 2. Concept used

In a horizontal ellipse, the length of the major axis is $2a$ and the length of the minor axis is $2b$.

In a vertical ellipse, the length of the major axis is$2b$ and length of the minor axis is $2a$.

Use the standard form of the equation of the ellipse and the formula is-

$c^2=b^2-a^2$

we will find the equation of an ellipse by substituting the values.

Step 3. Calculation

We are given Foci:$(0,\pm 2)$.

$\Rightarrow$ It is a vertical ellipse.

$\Rightarrow c=2$

Also, we are given that the length of the minor axis is $6$

$\begin{array}{l}\Rightarrow 2a=6\\ \Rightarrow a=3\\ \Rightarrow a^2=9\end{array}$

Using formula-

$\begin{array}{l}{c}^2=b^2-a^2\\ \Rightarrow 2^2=b^2-3^2\\ \Rightarrow 4=b^2-9\\ \Rightarrow b^2=9+4\\ \Rightarrow b^2=13\end{array}$

The equation of an ellipse is $\frac{x^2}{9}+\frac{y^2}{13}=1$