Question

question 17 of 20 which of the following correctly identifies the vertices that lie on the major axis of the conic section shown below?
\\(\frac{(x + 2)^2}{64}+\frac{(y - 1)^2}{81}=1\\)
a. (-10,1) and (6,1)
b. (-2,-7) and (-2,9)
c. (-11,1) and (7,1)
d. (-2,-8) and (-2,10)

Explanation:

Step1: Identify the form of the ellipse

The equation $\frac{(x + 2)^2}{64}+\frac{(y - 1)^2}{81}=1$ is of the form $\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1$ ($a>b>0$), where $(h,k)$ is the center of the ellipse. Here $h=-2,k = 1,a = 9,b = 8$.

Step2: Determine the vertices on the major axis

For an ellipse of the form $\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1$, the major axis is vertical and the vertices are given by $(h,k\pm a)$. Substituting $h=-2,k = 1,a = 9$ into the formula, we get the vertices as $(-2,1 + 9)$ and $(-2,1-9)$, which are $(-2,10)$ and $(-2,-8)$.

Answer:

D. (-2,-8) and (-2,10)