Find the standard form of the equation of the ellipse with vertices and eccentricity .

Question

Find the standard form of the equation of the ellipse with vertices  and eccentricity  .

Quizplus · Accepted Answer

From the given information, we can first find that the center of the ellipse is the midpoint of the vertices, which is $(0,0)$. Since the eccentricity is greater than $1$, we know that the ellipse is a stretched out horizontally. Using the formula $a^2=c^2-b^2$, where $a$ is the semi-major axis, $b$ is the semi-minor axis and $c$ is the distance from the center to each vertex, we can find $a$:
\begin{align*}
a^2&=c^2-b^2\
a^2&=(11eacce6\_20f8\_4fb9\_9f49\_bf6c8381cd66\_TB7896\_11/2)^2-(20f8-4fb8-9f49-73f89307e081\_TB7896\_11/2)^2\
a^2&=(11eacce6\_20f8\_4fb9\_9f49\_bf6c8381cd66\_TB7896\_11/2)^2-(-20f8+4fb8+9f49+73f89307e081\_TB7896\_11/2)^2\
a^2&=76cc-9f49-f5085f405b73\_TB7896\_16\
\end{align*}
Therefore, the equation of the ellipse in standard form is:
$$\frac{x^2}{76cc-9f49-f5085f405b73\_TB7896\_16}+\frac{y^2}{(\sqrt{76cc-9f49-f5085f405b73\_TB7896\_16}\cdot11eacce6\_20f8\_4fb9\_9f49\_bf6c8381cd66\_TB7896\_11)^2}=1$$
which simplifies to $\boxed{	extbf{(C)}}$.

Find the Standard Form of the Equation of the Ellipse