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If an ellipse has an equation in the standard form and it passes through the points ( 5 2 , 6 4 ) and (-2, 15 5 ) then the length of its latus rectum is

Options

  1. A10 5
  2. B10 5
  3. C1 10
  4. D1 10

Correct answer

A. 10 5

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 . Substituting the point ( 5 2 , 6 4 ) into the equation: 25 4a^2 + 6 16b^2 = 1 25 4a^2 + 3 8b^2 = 1 . Substituting the point (-2, 15 5 ) into the equation: 4 a^2 + 15 25b^2 = 1 4 a^2 + 3 5b^2 = 1 . Let X = 1 a^2 and Y = 1 b^2 . The equations become: 25 4 X + 3 8 Y = 1 and 4X + 3 5 Y = 1 . Multiplying the first by 8 : 50X + 3Y = 8 . Multiplying the second by 5 : 20X + 3Y = 5 . Subtracting the two equations: 30X = 3 X = 1 10 = 1 a^2 a^2 = 10 . Substituting X i

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