Question

Write an equation of an ellipse centered at the orgin, satisfying the given conditions.
Foci (+-2,0) ; Co vertices (0, +-6)
Medal will be given. (:

Answer 1

The equation for an ellipse with the center at the origin is \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \]
The a represents the distance from the vertices to the center of the ellipse, and the b represents the distance from the co-vertices to the center of the ellipse.
Since we are not given the vertices here, we have to use a^2-b^2=c^2 to find them, in this case, c represents the foci.
So we have:
\[a^2-b^2=c^2\]
\[a^2-(6)^2=2^2\]\[a^2=4+36\]\[a=-\sqrt{40} and +\sqrt{40}\]
Now you have all the components you need to make the equation for this ellipse.
\[\frac{x^{2}}{ (\sqrt{40})^{2}}+\frac{y^{2}}{6^{2}}=1 \]The equation of this ellipse is:
\[\frac{x^{2}}{40}+\frac{y^{2}}{36}=1 \]