Question

Finding the Equation of an Ellipse

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

Length of major axis: $4$, length of minor axis: $2$, foci on $y$-axis

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 as $4$, length of minor axis as 2 and foci on y axis are given

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 y-axis.

$\Rightarrow$ It is a vertical ellipse.

We are given the length of the major axis is 4 and the length of the minor axis is 2 .

$\Rightarrow 2b=4$ and

$\begin{array}{l}\Rightarrow b=2\\ \Rightarrow b^2=4\\ \Rightarrow 2a=2\\ \Rightarrow a=1\\ \Rightarrow a^2=1\end{array}$

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