Question 11.3.3: Finding the Equation of an Ellipse Find an equation of the e...

Because the foci (-3, 2) and (5, 2) lie on the horizontal line y = 2 the ellipse is a horizontal ellipse. The center of the ellipse is the midpoint of the line segment joining the foci. Using the midpoint formula yields

h=\frac{-3+5}{2}=1 \quad \text { and } \quad k=\frac{2+2}{2}=2.

The center of the ellipse is (1, 2) See Figure 17. Because the length of the major axis is 10, the vertices must be at a distance a = 5 units from the center.
In Figure 17, the foci are four units from the center. Thus, c=4 . \text { Now find } b^{2} :

b^{2}=a^{2}-c^{2}=(5)^{2}-(4)^{2}=9

Because the major axis is horizontal, the standard form of the equation of the ellipse is

\begin{array}{ll}\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 & a>b>0 \\\frac{(x-1)^{2}}{25}+\frac{(y-2)^{2}}{9}=1 & \text { Replace } h \text { with } 1, k \text { with } 2, \\ &a^{2} \text { with } 25, \text { and } b^{2} \text { with } 9 .\end{array}

For this horizontal ellipse with center (1,2) \text {, we have } a=5, b=3 \text {, and } c=4 \text {. }

Vertices:

(h \pm a, k)=(1 \pm 5,2)=(-4,2) \text { and }(6,2)

Endpoints of Minor Axis:

(h, k \pm b)=(1,2 \pm 3)=(1,-1) \text { and }(1,5)

Using the center and these four points we sketch the graph shown in Figure 18.