Write the equation of the ellipse that has its center at the origin, focus at (2, 0) and vertex at (7, 0).

Question

Quizplus · Accepted Answer

Since the center of the ellipse is at the origin, the equation must take the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively. We are given that the focus is at (2,0), which means the distance from the origin to the focus is $c=2$. We also know that the vertex is at (7,0), which means the distance from the center to the vertex is $a=7$.

Recall that the distance between the center and each focus is given by $c=\sqrt{a^2-b^2}$. Using the values we have, we can solve for $b$ as follows:

$$c=\sqrt{a^2-b^2}$$
$$2 = \sqrt{49-b^2}$$
$$4 = 49-b^2$$
$$b^2 = 45$$

Therefore, the equation of the ellipse is $\frac{x^2}{49}+\frac{y^2}{45}=1$. This matches with option B.

Write the Equation of the Ellipse That Has Its Center