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The endpoints of the latus rectum of a parabola are L₁(-2, 10) and L₂(14, -2) . If the x -coordinate of the vertex of this parabola is less than 5 , and its equation is 16x^2 + py^2 + qxy + rx + sy + t = 0 , then the value of p + q + r + s + t is equal to

Options

  1. A-5095
  2. B-1870
  3. C720
  4. D305

Correct answer

D. 305

Step-by-step solution

The focus S of the parabola is the midpoint of the latus rectum L₁L₂ . S = ( -2 + 14 2 , 10 - 2 2 ) = (6, 4) . The length of the latus rectum is 4a = (14 - (-2))^2 + (-2 - 10)^2 = 16^2 + (-12)^2 = 20 . Thus, a = 5 . The axis of the parabola is the perpendicular bisector of L₁L₂ . Slope of L₁L₂ = -2 - 10 14 - (-2) = - 12 16 = - 3 4 . Slope of the axis = 4 3 . Equation of the axis passing through S(6, 4) : y - 4 = 4 3 (x - 6) 4x - 3y - 12 = 0 . The vertex V lies on the axis at a distance a = 5 from the focus S(6, 4)

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