Question

Write an equation for an ellipse centered at the origin, which has foci at ( 0 , ± 24 ) (0,±24)left parenthesis, 0, comma, plus minus, 24, right parenthesis and co-vertices at ( ± 10 , 0 ) (±10,0)

Answer 1

Answer:

$\frac{x^{2}}{400} + \frac{y^{2}}{976} = 1$

Step-by-step explanation:

The distance between foci with respect to origin is determined by mean of the Pythagorean Theorem:

$2\cdot c = \sqrt{(0-0)^{2}+[24-(-24)]^{2}}$

$2\cdot c = 48$

$c = 24$

The distance between origin and any of the horizontal co-vertices is:

$a = \sqrt{[10-(-10)]^{2}+(0-0)^{2}}$

$a = 20$

Now, the distance between origin and any of the vertical co-vertices is determined by the following Pythagorean relationship:

$c^{2} = b^{2} - a^{2}$

$b^{2} = a^{2} + c^{2}$

$b = \sqrt{a^{2}+c^{2}}$

$b = \sqrt{20^{2}+ 24^{2}}$

$b = 4\sqrt{61}$

Lastly, the equation of the ellipse in standard form is:

$\frac{x^{2}}{400} + \frac{y^{2}}{976} = 1$