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An ellipse x^2 a^2 + y^2 b^2 = 1 ( a > b ) meets the x -axis at the same point where the straight line 3x + 4y = 12 meets the positive x -axis. If the line y = x + 2 6 is a tangent to this ellipse, then the length of its latus rectum is

Options

  1. A18 15
  2. B10
  3. C4
  4. D2

Correct answer

C. 4

Step-by-step solution

The line 3x + 4y = 12 meets the x -axis where y = 0 . Substituting y = 0 , we get 3x = 12 x = 4 . Since the ellipse x^2 a^2 + y^2 b^2 = 1 meets the positive x -axis at this same point, its semi-major axis is a = 4 . The condition for a line y = mx + c to be tangent to the ellipse is c^2 = a^2m^2 + b^2 . For the given tangent line y = x + 2 6 , we have m = 1 and c = 2 6 . Substituting the known values into the tangency condition: (2 6 )^2 = (4)^2(1)^2 + b^2 24 = 16 + b^2 b^2 = 8 The length of the latus rectum of the

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