Question

Foci of an ellipse are at S(1. 7). S'(1,-3). The point is on the ellipse such that SP = 7, S'P = 5 Then the equation of the ellipse is​

Answer 1

Foci of an ellipse are at S(1, 7) and S'(1, -3). The point is on the ellipse such that SP = 7 and S'P = 5.

To find : The equation of the ellipse.

solution : using concept of ellipse,

SP + S'P = SS'/e

SP = 7, S'P = 5 and SS' = √{(1 - 1)² + (7 + 3)²} = 10

so 7 + 5 = 12 = 10/e

⇒e = 10/12 = 5/6

and length of major axis = 2a = SP + S'P

⇒2a = 7 + 5

⇒ a = (7 + 5)/2 = 6

using formula, b² = a²(1 - e²)

⇒b² = 6²(1 - 25/36) = 36 × (36 - 25)/36 = 11

⇒b = ±√11

C is the midpoint of S and S'

i.e., C = [(1 + 1)/2, (7 - 3)/2 ] = (1, 2)

now equation of ellipse is given by,

⇒ (x - 1)²/6² + (y - 2)²/(±√11)² = 1

⇒(x - 1)²/36 + (y - 2)²/11 = 1

Therefore the equation of the ellipse is given by, (x - 1)²/36 + (y - 2)²/11 = 1

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