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Question

The tangent PT and the normal PN to the parabola y2=4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola, whose
  1. directrix is x = 0
  2. vertex is (2a3,0)
  3. focus is (a, 0)
  4. latus rectum is 2a3


Solution

The correct option is C focus is (a, 0)
Equation of tangent and normal at point P(at2,2at) is 
ty=x+at2 and y=tx+2at+at3
Let centroid of ΔPTN is R(h,k).
h=at2+(at2)+2a+at23


and k=2at33h=2a+a(3k2a)2
3h=2a+9k24a
9k2=4a(3h2a)
Locus of centroid is
y2=4a3(x2a3)
Vertex (2a3,0); directrix
x2a3=a3x=a3
and latus rectum =4a3
Focus (a3+2a3,0),i.e.(a,0).
 

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