Answer:
B: ((x - 3)^2)/33 + ((y - 2)^2)/49 = 1
Explanation:
To find the standard equation of the ellipse, we can use the formula for the distance between a point (x, y) and a focus (h, k) on the ellipse:
√((x - h)^2 + (y - k)^2)
Given that the foci are (-3, -6) and (-3, 2) and the sum of distances from a point on the ellipse to the foci is 14, we have:
√((x - (-3))^2 + (y - (-6))^2) + √((x - (-3))^2 + (y - 2)^2) = 14
Simplifying the equation, we have:
√((x + 3)^2 + (y + 6)^2) + √((x + 3)^2 + (y - 2)^2) = 14
To put this equation in standard form, we need to isolate one of the square root terms on one side:
√((x + 3)^2 + (y + 6)^2) = 14 - √((x + 3)^2 + (y - 2)^2)
Now, let's square both sides to eliminate the square root:
((x + 3)^2 + (y + 6)^2) = (14 - √((x + 3)^2 + (y - 2)^2))^2
Expanding the squared term on the right side:
(x + 3)^2 + (y + 6)^2 = (14 - √((x + 3)^2 + (y - 2)^2))(14 - √((x + 3)^2 + (y - 2)^2))
Multiplying out the right side using the distributive property:
(x + 3)^2 + (y + 6)^2 = 196 - 14√((x + 3)^2 + (y - 2)^2) - 14√((x + 3)^2 + (y - 2)^2) + ((x + 3)^2 + (y - 2)^2)
Now, let's simplify the equation:
(x + 3)^2 + (y + 6)^2 = 196 - 28√((x + 3)^2 + (y - 2)^2) + ((x + 3)^2 + (y - 2)^2)
Rearranging terms:
2(x + 3)^2 + 2(y - 2)^2 + 28√((x + 3)^2 + (y - 2)^2) = 196
Dividing both sides by 196:
[(x + 3)^2]/98 + [(y - 2)^2]/49 = 1
Therefore, the standard equation of the ellipse is option B:
((x - 3)^2)/33 + ((y - 2)^2)/49 = 1