Question

Finding the Equation of an Ellipse

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

End points of minor axis: $(0,\pm 3)$, distance between foci is: $8$

Short answer

The equation of ellipse is $\frac{x^2}{25}+\frac{y^2}{9}=1$

Step-by-step solution

Step 1. Given information

End points of minor axis $(0,\pm 3)$and the distance between foci is $8$.

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 end points of minor axis as: $(0,\pm 3)$

$\Rightarrow$ It is a horizontal ellipse.

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

Also, we are given that the distance between foci is $8$

$\begin{array}{l}\Rightarrow 2c=8\\ \Rightarrow c=4\end{array}$

Using formula-

$\begin{array}{l}{c}^2=a^2-b^2\\ \Rightarrow 4^2=a^2-3^2\\ \Rightarrow 16=a^2-9\\ \Rightarrow 16+9=a^2\\ \Rightarrow a^2=25\end{array}$

The equation of an ellipse is $\frac{x^2}{25}+\frac{y^2}{9}=1$