Question

The tangent PT and the normal PN to the parabola `y^2=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:
A. vertex is `((2a)/(3),0)`
B. directrix is x = 0
C. latusrectum is `(2a)/(3)`
D. focus is (a,0)

Answer 1

Correct Answer - A::D
Equation of tangent and normal at point `P(at^(2), 2at)` is
`ty = x = at^(2)` and `y = - tx + 2at + at^(2)`
Let controid of `Delta PTN` is R(h,k)
`:. H = (at^(2) + (-at^(2)) + 2a + at^(2))/(3)`

and `k = (2at)/(3)`
`implies 3h = 2a + a ((3k)/(2a))^(2)`
`implies 3h = 2a + (9k^(2))/(4a)`
`implies 9k^(2) = 4a (3h - 2a)`
`:.` Locus of centroid is
`y^(2) = (4a)/(3) (x - (2a)/(3))`
`:.` vertex `((2a)/(3), 0)`, directrix
`x - (2a)/(3) = - (a)/(3)`
`implies x = (a)/(3)`
and latusrectum `= (4a)/(3)`
`:.` Focus `((a)/(3) + (2a)/(3), 0)` i.e., (a,0)