Chapter 8.2, Problem 2E

Solution

To calculate: The vertices and foci of the ellipse.

The vertices of the ellipse are $\left(5,0\right)$ , $\left(-5,0\right)$ and the foci are $\left(2,0\right),\left(-2,0\right)$ .

Given information:

The equation of ellipse is $\frac{x^2}{25}+\frac{y^2}{21}=1$ .

Calculation:

The vertices are the two points on the ellipse that intersect the major axis. For an ellipse with major axis parallel to the $x-$ axis, the vertices are $\left(h+a,k\right),\left(h-a,k\right)$ .

  $\frac{x^2}{25}+\frac{y^2}{21}=1$

Compare the equation with standard equation of ellipse:

Ellipse with center $\left(h,k\right)=\left(0,0\right)$ .

  $\begin{array}{l}a=\sqrt{25}=5\\ b=\sqrt{21}\end{array}$

Thus, the vertices of the ellipse are $\left(5,0\right)$ , $\left(-5,0\right)$ .

Calculate the value of $c:$

  $\begin{array}{l}c=\sqrt{5^2-{\left(\sqrt{21}\right)}^2}\\ c=\sqrt{25-21}\\ c=\sqrt{4}\\ c=2\end{array}$

The foci of the ellipse are defined as $\left(h+c,k\right),\left(h-c,k\right)$ .

Thus, the foci are $\left(2,0\right),\left(-2,0\right)$ .