Answer to Question #143611 in Analytic Geometry for Sasai

Question

Question 1100429: Which of the following equations represents an ellipse having a major axis of length 18 and foci located at 4,7) and 4,11)? (There's supposed to be a parenthesis there I can't put it something pops up.) I'm doing all this on my computer there really hasent been any examples on how to do these and there's no teacher lecturing all this which is why I need help with all these.
A.(x-9)^2/81+(y-4)^2/4=1
B.(x-4)^2/77+(y-9)^2/81=1
C.(x-9)^2/77+(y-4)^2/81=1
D.(x-4)^2/81+(y-9)^2/77=1
I don't think it's D or A

Answer 1

The center of the ellipse lies in the middle of the segment connecting the foci. Let's define it's coordinates.

$c = ( (F1_x + F2_x)/2, (F1_y + F2_y)/2 ) = ( (4 + 4)/2, (7 + 11)/2 ) = (4, 9)$

where (4, 7), (4, 11) - given foci coordinates

This leaves us with options

B. $(x-4)^2/77+(y-9)^2/81=1$

and

D. $(x-4)^2/81+(y-9)^2/77=1$

As the foci have the same x coordinate therefore major axis is parallel to the y-coordinate axis and squared semi-major axis length must be the divisor of the y-coordinate in the ellipse equation.

Squared semi-major axis length: $a^2 = ( 18 /2 )^2 = 81$

where 18 - given major axis length

This leaves us with option

B. $(x-4)^2/77+(y-9)^2/81=1$

Answer: B.