Question

Finding the Equation of an Ellipse

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

Foci: $(\pm 5,0)$, length of major axis: $12$

Short answer

The equation of ellipse is $\frac{x^2}{36}+\frac{y^2}{11}=1$

Step-by-step solution

Step 1. Given information

Foci $(\pm 5,0)$The length of the major axis is $12$.

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: $(\pm 5,0)$.

$\Rightarrow$ It is a horizontal ellipse.

$\Rightarrow c=5$

Also, we are given that the length of the major axis is $12$

$\begin{array}{l}\Rightarrow 2a=12\\ \Rightarrow a=6\end{array}$

Using formula-

$\begin{array}{l}{c}^2=a^2-b^2\\ \Rightarrow 5^2=6^2-b^2\\ \Rightarrow 25=36-b^2\\ \Rightarrow b^2=36-25\\ \Rightarrow b^2=11\end{array}$

The equation of an ellipse is $\frac{x^2}{36}+\frac{y^2}{11}=1$