Write the equation of the ellipse that has its center at the origin with focus at (0, 2) and vertex at (0, 10).
E)none of these

Question

Quizplus · Accepted Answer

Since the center is at the origin, we have $h = 0$ and $k = 0$.

Since the distance from the center to the focus is 2, we have $c = 2$.

Since the distance from the center to the vertex is 10, we have $a = 10$.

Recall that the standard form of the equation of an ellipse with a horizontal major axis is:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

where $a$ is the distance from the center to the vertex (in our case, 10), and $b$ is the distance from the center to the co-vertex. We know that $c^2 = a^2 - b^2$, so:

$$2^2 = 10^2 - b^2 \quad\Rightarrow\quad b^2 = 96$$

Therefore, the equation of the ellipse is:

$$\frac{x^2}{10^2} + \frac{y^2}{96} = 1$$

Multiplying both sides by 96, we get:

$$\frac{96x^2}{100} + y^2 = 96$$

Simplifying and rearranging, we get:

$$\frac{x^2}{\frac{100}{96}} + \frac{y^2}{96} = 1$$

So the final equation is:

$$\frac{x^2}{\frac{25}{24}} + \frac{y^2}{96} = 1$$

which matches the equation in choice B.

Write the Equation of the Ellipse That Has Its Center