Question

Find an equation of the ellipse having a major axis of length 10 and foci at , (1,−2 )and , (−7,−2.)

Answer 1

The equation of an ellipse is

$$\frac{\left(x - h\right)^{2}}{a^{2}} + \frac{\left(y - k\right)^{2}}{b^{2}} = 1,$$

where $\left(h, k\right)$ is the center, $a$ and $b$ are the lengths of the semi-major and the semi-minor axes.

Thus $a=10/2=5.$

From properties of an ellipse

$$k=-2$$

$$(h-1)^2=a^2-b^2$$

$$(h+7)^2=a^2-b^2$$

Then

$$(h-1)^2=(h+7)^2$$

$$h^2-2h+1=h^2+14h+49$$

$$16h=-48$$

$$h=-3$$

$$(-3-1)^2=5^2-b^2$$

$$b^2=9$$

The equation of the ellipse is

$$\frac{\left(x+3\right)^{2}}{25} + \frac{\left(y +2\right)^{2}}{9} = 1,$$