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AP EAMCET2010MathematicsParabola

Let M be the foot of the perpendicular from a point P on the parabola y^2=8(x-3) onto its directrix and let S be the focus of the parabola. If S P M is an equilateral triangle, then P is equal to

Options

  1. A(4 3 , 8)
  2. B(8,4 3 )
  3. C(9,4 3 )
  4. D(4 3 , 9)

Correct answer

C. (9,4 3 )

Step-by-step solution

Given, that the S P M (which is shown in figure) is equilateral. Also, given parabola is y^2=8(x-3) focus of this parabola is S(5,0) and vertex A(3,0) . Let coordinate of P (h+a t^2, k+2 a t )=P (3+2 t^2, 4 t ) Then, coordinate of M(-5,4 t) . We know that the side of this equilateral triangle is 4 a=4 2=8 Now, P S=8 (3+2 t^2-5 )^2+(4 t)^2 =8 (2 t^2-2 )^2+(4 t)^2 =8 (2 t^2+2 )^2 =8 2 t^2+2=8 2 t^2=6 t= 3 P(3+2 3,4 3 )=P(9,4 3 )

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