Question

Which points are the foci of the ellipse?

A. (−8, −1) and (−2, −1)
B. (−6, −1) and (−5, −1)
C. (−5, −4) and (−5, 2)
D. (−5, −2) and (−5, 0)

Answer 1

Answer: C. (−5, −4) and (−5, 2)

Explanation:

Answer 2

check the picture below.

notice, the longer axis is the vertical one, therefore, "a" component is under the "y" variable expression, and therefore, the focus points, will move out of the center up and down.

also notice, the center is at -5, -1.


`\bf \textit{ellipse, vertical major axis} \\\\ \cfrac{(x- h)^2}{ b^2}+\cfrac{(y- k)^2}{ a^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad √( a ^2- b ^2)\\ ----------\\ a=5\\ b=4\\ \end{cases} \\\\\\ c=√(5^2-4^2)\implies c=√(9)\implies c=\pm 3 \\\\\\ \textit{therefore the foci are at}\qquad (h~,~k\pm c)\implies (-5~~,~~-1 \pm 3)`