Question

Finding the Equation of an Ellipse

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

Eccentricity: $\frac{\sqrt{3}}{2}$, Foci on y-axis and length of major axis is 4

Short answer

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

Step-by-step solution

Step 1. Given information

Length of major axis 4 and the eccentricity is $\frac{\sqrt{3}}{2}$.

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

It is given that, length of the major axis is 4

Since the foci are on the y axis then the major axis is also along the y axis.

Therefore, the equation of the ellipse will be of the form $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

where $a>b$

The length of the major axis is 4

$\begin{array}{l}2a=4\\ a=2\end{array}$

Squaring both sides we get

$a^2=4$

Eccentricity is $\frac{\sqrt{3}}{2}$

We know that $b^2=a^2\left(1-e^2\right)$

$\begin{array}{l}{b}^2=4\left(1-{\left(\frac{\sqrt{3}}{2}\right)}^2\right)\\ b^2=4\left(1-\frac{3}{4}\right)\\ {\text{b}}^2=4\left(\frac{1}{4}\right)\\ b^2=1\end{array}$

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