Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.
Vertices: ; Foci:

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Vertices:  ; Foci:  ​

Quizplus · Accepted Answer

Since the center is at the origin, we know that the equation will have the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

The distance between the vertices is $2a$, so we can find $a$ by computing the distance between the given vertices, which is $\sqrt{\left(11eacce6\_20e6\_7587\_9f49\_957d24324230\right)^2+\left(11eacce6\_20e6\_c3ac\_9f49\_cf9f0bd9fbd1\right)^2}$. Similarly, the distance between the foci is $2c$, where $c$ is the distance from the center to each focus. We can find $c$ using the distance formula with the given foci.

Using these values, we can solve for $b$ in the equation $\frac{c^2}{a^2}-\frac{b^2}{a^2}=1$. Once we have $a$ and $b$, we can plug them into the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to get the standard form of the equation.

After computing, we get that $a=3$ and $b=2\sqrt{2}$, and plugging these into the equation yields $\frac{x^2}{9}+\frac{y^2}{8}=1$. Therefore, the answer is choice C.

Find the Standard Form of the Equation of the Ellipse