Find the equation in standard form of an ellipse with foci that passes through the point .

Question

Find the equation in standard form of an ellipse with foci  that passes through the point  .

Quizplus · Accepted Answer

Since the ellipse passes through one of its foci, the center of the ellipse must lie on the perpendicular bisector of the segment connecting the given point and the focus.
Let the center of the ellipse be (h, k), and let the distance from the center to each focus be c. Then, the distance from the center to the given point is also c. 
Using the distance formula and the fact that c = 6, we have:
√((h-11)^2 + k^2) + √((h+11)^2 + k^2) = 2c
√((h-11)^2 + k^2) + √((h+11)^2 + k^2) = 12
Squaring both sides and simplifying, we get:
h^2/9 + k^2/36 = 1
The standard form of an ellipse with center (0, 0) is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the major and minor axes, respectively. 
To put the given equation in standard form, we need to divide both sides by 4 and rearrange:
(h^2/36) + (k^2/9) = 1
Therefore, the equation in standard form is 9x^2 + 36y^2 = 324, which corresponds to choice D.

Find the Equation in Standard Form of an Ellipse with Foci