10. Which of the following is an equation of the ellipse with center (-2,1), major axis running from (-2,6) to (-2,-4), and focus at (-2,5)?
A. (x-2)^{2}/25+(y+1)^{2}/16=1 |
B. (x+2)^{2}/16+(y-1)^{2}/25=1 |
C. (x-2)^{2}/9+(y+1)^{2}/25=1 |
D. (x+2)^{2}/9+(y-1)^{2}/25=1 |
E. (x-2)^{2}/25+(y-1)^{2}/9=1 |
Solution: The answer is D
Let 2a be the length of the major axis and 2b be the length of the minor axis where a > b > 0. Also, let c be the distance between the foci and the center of the ellipse, where c^{2} = a^{2} – b^{2}.
Since the major axis runs from (-2,6) to (-2,-4), then the major axis has a vertical length of,
Since the center is at (-2,1) and the focus is at (-2,5), then the distance c between the focus and the center is,
So, that b^{2} is given by,
Lastly, since the standard form of the equation of an ellipse with a vertical major axis is given by,
where (h,k) is the center, then the equation of the ellipse represented by the given data is,