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Let the circle 2x^2 + 2y^2 - 15x - 10y = 0 intersect the x -axis and y -axis at the points P and Q , respectively, where neither P nor Q is the origin. If the line passing through P and Q is a tangent to the ellipse x^2 a^2 + y^2 9 = 1 (where a > 3 ), then the length of the latus rectum of this ellipse is

Options

  1. A3
  2. B24
  3. C6
  4. D9

Correct answer

A. 3

Step-by-step solution

To find the coordinates of P and Q , we determine the non-zero intercepts of the circle 2x^2 + 2y^2 - 15x - 10y = 0 on the coordinate axes. For the x -intercept, put y = 0 : 2x^2 - 15x = 0 x(2x - 15) = 0 . Since P is not the origin, x = 15 2 , so P ( 15 2 , 0 ) . For the y -intercept, put x = 0 : 2y^2 - 10y = 0 y(2y - 10) = 0 . Since Q is not the origin, y = 5 , so Q (0, 5) . The equation of the line passing through P and Q in intercept form is: x 15/2 + y 5 = 1 2x 15 + y 5 = 1 2x + 3y = 15 . Rewriting this in slop

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