Chapter 9, Problem 11CR

To determine

To find the focus of the ellipse $\frac{x^2}{4}+\frac{y^2}{20}=1$ .

Answer to Problem 11CR

The Focus is $a.$

  $\left(0,4\right)$ .

Explanation of Solution

Given:

Equation of Ellipse: $\frac{x^2}{4}+\frac{y^2}{20}=1$

Formula Used:

The standard equation for Ellipse - $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ with Center as $\left(h,k\right)$ and foci as $\left(h,k+c\right),\left(h,k-c\right)$ where $c=\sqrt{b^2-a^2}$

Calculation:

Given:

Equation of Ellipse: $\frac{x^2}{4}+\frac{y^2}{20}=1$

The equation can be simplified as:

  $\frac{x^2}{2^2}+\frac{y^2}{{\left(2\sqrt{5}\right)}^2}=1$

Using the above formula and comparing the above values in the equation of the Ellipse, we have:

  $h=0$

And $k=0$

Thus,

Center $=$$\left(h,k\right)$

Center $=$$\left(0,0\right)$

  $a=2$ and $b=2\sqrt{5}$

  $c=\sqrt{b^2-a^2}$

  $\begin{array}{l}c=\sqrt{20-4}\\ c=4\end{array}$

Now, Foci $=$$\left(h,k+c\right),\left(h,k-c\right)$

Foci $=$$\left(0,0+4\right),\left(0,0-4\right)$

Foci $=$$\left(0,4\right),\left(0,-4\right)$

Thus, one of the Focus is $a.$

  $\left(0,4\right)$

Conclusion:

The Focus is $a.$

  $\left(0,4\right)$ .