Question

Finding the Equation of an Ellipse

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

Length of major axis: 6, length of minor axis: 4, foci on x-axis

Short answer

The equation of ellipse is $\frac{x^2}{9}+\frac{y^2}{4}=1$

Step-by-step solution

Step 1. Given information

The length of the major axis is 6. The length of the minor axis is 4. Foci are on x-axis.

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 on x-axis.

$\Rightarrow$It is a horizontal ellipse.

We are given the length of the major axis is 6 and the length of the minor axis is 4.

$\Rightarrow 2a=6$and

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

The equation of an ellipse is $\frac{x^2}{9}+\frac{y^2}{4}=1$