Write the equation of the ellipse that has its center at the origin with focus at (0, 5) and vertex at (0, 10).
E)none of these

Question

Quizplus · Accepted Answer

Since the center is at the origin, we know that the equation of the ellipse is of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We also know that the distance from the center to the focus is $c=5$, and the distance from the center to the vertex is $a=10$. We can use the formula $c^2=a^2-b^2$ to solve for $b$:
$$5^2=10^2-b^2 \Rightarrow b^2=75$$
Plugging in $a=10$ and $b=\sqrt{75}$, we get the equation:
$$\frac{x^2}{10^2}+\frac{y^2}{\sqrt{75}^2}=1$$
Simplifying and rationalizing the denominator, we get the equation:
$$\frac{x^2}{100}+\frac{y^2}{25}=1$$
So the answer is B.

Write the Equation of the Ellipse That Has Its Center