Question

Write an equation for an ellipse centered at the orgin, which has foci ( plus minus 3, 0) and co-vertices at (0, plus minus 4)

Answer 1

Answer:

The answer is below

Step-by-step explanation:

The co-vertices of an ellipse are the endpoints of the minor axis. The equation for an ellipse is given by:

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

Where (h, k) is the center of the ellipse, (h, k±b) is the co-vertices, (h ± a, k) is the vertices, (h ± c, k) is the foci and c² = a² - b²

Since the center is the origin, hence (h, k) = (0, 0). i.e h = 0, k = 0.

Foci = (h ± c, k) = (± c, 0) = (±3, 0). c = 3

co-vertices = (h, k±b)  = (0, ±b)  = (0, ±4). b = 4

c² = a² - b²

a² = c² + b²

a² = 3² + 4² = 25

a = 5

Therefore the equation of the ellipse is:

$\frac{x^2}{5^2} +\frac{y^2}{4^2}=1 \\\\\frac{x^2}{25} +\frac{y^2}{16}=1$