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The foci of an ellipse are (-3,-6) and (-3,2). For any point on the ellipse, the sum of its distances from the foci is 14 . Find the standard equation of the ellipse. A. ((x+3)^(2))/(33)+((y+2)^(2))/(49)=1 B. ((x-3)^(2))/(33)+((y-2)^(2))/(49)=1 C. ((x+3)^(2))/(49)+((y+2)^(2))/(13)=1 D. ((x-3)^(2))/(49)+((y-2)^(2))/(33)=1

User Ted
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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

User AminSojoudi
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