The length of the latus rectum of the ellipse $9(2x + 3y - 1)^{2} + 16(3x - 2y + 5)^{2} = 25$ is:
Ans. (c)

Question

The length of the latus rectum of the ellipse $9(2x + 3y - 1)^{2} + 16(3x - 2y + 5)^{2} = 25$ is:Ans. (c)

Accepted Answer

Given, 9(2x + 3y − 1)² + 16(3x − 2y + 5)² = 25

Since (2, 3) ⟂ (3, −2), the transformation is only a rotation and translation.

Let X = (2x + 3y − 1)/√13, Y = (3x − 2y + 5)/√13

Then, 117X² + 208Y² = 25

Or X²/(25/117) + Y²/(25/208) = 1

Thus, a² = 25/117, b² = 25/208

Since a² > b², the length of latus rectum is

L = 2b²/a = 2(25/208)/(5/√117) = 5√117/104

Therefore,

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