Question

find the equation in standard form for each the ellipse with vertices at (2, 9) and (2, -1) and minor axis of length 10.

Explanation:

Step1: Find the center of the ellipse

The center of the ellipse is the mid - point of the vertices. The vertices are $(2,9)$ and $(2, - 1)$. The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1=x_2 = 2$, $y_1=9$ and $y_2=-1$. So the center $(h,k)=(2,\frac{9+( - 1)}{2})=(2,4)$.

Step2: Determine the values of $a$ and $b$

The distance between the vertices is the length of the major axis. The distance between $(2,9)$ and $(2,-1)$ is $d=\vert9-( - 1)\vert = 10$, so the length of the major axis $2a = 10$, then $a = 5$. The length of the minor axis is given as $10$, so $2b=10$, then $b = 5$.

Step3: Write the standard form of the ellipse equation

Since the vertices have the same $x$ - coordinate, the major axis is vertical. The standard form of the equation of an ellipse with a vertical major axis is $\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1$. Substituting $h = 2,k = 4,a = 5,b = 5$ into the equation, we get $\frac{(x - 2)^2}{25}+\frac{(y - 4)^2}{25}=1$.

Answer:

$\frac{(x - 2)^2}{25}+\frac{(y - 4)^2}{25}=1$