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 ΔPTN is a parabola whose

(a)  vertex is(2a/3,0)

(B)  direction is x = 0

(C)  latus rectum is 2a/3

(D)  focus is (a, 0) 

Answer 1

Correct option   (A), (D)

Explanation :

See Fig. P = (at², 2at). Tangent at P is

ty = x + at²

 so that T = (-at², 0). Normal at P is

 tx + y = 2at + at³

 and hence N = (2a + at², 0)

 Now P = (at², 2at), T = (-at², 0) and N = (2a + at², 0). Let G(x, y) be the centroid of ΔPTN. Therefore

x = 2a + at²/3  and y = 2at/3

 Hence

 Therefore