Question

Finding the Equation of an Ellipse

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

Foci: $F(0,\pm \sqrt{10})$, vertices: $(0,\pm 7)$

Short answer

the equation of ellipse is $\frac{x^2}{39}+\frac{y^2}{49}=1$

Step-by-step solution

Step 1. Given information

Foci $F(0,\pm \sqrt{10})$ and vertices $(0,\pm 7)$ are given

Step 2. Concept used

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: $(0,\pm \sqrt{10})$ and vertices: $(0,\pm 7)$

$\begin{array}{l}\Rightarrow b=7\\ \Rightarrow c=\sqrt{10}\end{array}$

Thus, the major axis is vertical.

We will find $a^2$ from $c^2=b^2-a^2$

Substituting $b=7$and

$c=\sqrt{10}$ and simplifying

$\Rightarrow {(}^{\sqrt{10})2}=7^2-a^2$

$\begin{array}{l}\Rightarrow 10=49-a^2\\ \Rightarrow a^2=49-10\\ \Rightarrow a^2=39\end{array}$

The equation of an ellipse is $\frac{x^2}{39}+\frac{y^2}{49}=1$