Question

homework 10: ellipse and hyperbola
score: 5/20 answered: 5/20
question 6
given the ellipse \\( \frac{(x - 5)^{2}}{4}+\frac{(y - 6)^{2}}{16}=1 \\)
find the center point:
list the two major vertices (separated by a comma):
question help: video 1 video 2

Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse is $\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1$ ($a>b$), where $(h,k)$ is the center.
For the given ellipse $\frac{(x - 5)^2}{4}+\frac{(y - 6)^2}{16}=1$, we have $h = 5$ and $k=6$.

Step2: Find the major vertices

Since $a^2 = 16$, then $a = 4$. The major axis is vertical (because the denominator of the $y$-term is larger).
The major vertices are $(h,k\pm a)$. Substituting $h = 5,k = 6,a = 4$, we get $(5,6 + 4)$ and $(5,6-4)$.

Answer:

Center point: $(5,6)$
Major vertices: $(5,10),(5,2)$