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 $(\frac{2a}{3},0)$
• B. directrix is x = 0
• C. latus rectum is $\frac{2a}{3}$
• D. focus is (a, 0)

Answer 1

Answer: (A), (D)

focus is (a, 0)

Equation of tangent and normal at point $P(a{t}^2,2at)$ is
$ty=x+a{t}^2$ and $y=-tx+2at+a{t}^3$
Let centroid of $\Delta PTN$ is $R(h,k)$.
$\therefore h=\frac{a{t}^2+(-a{t}^2)+2a+a{t}^2}{3}$


and $k=\frac{2at}{3}\Rightarrow 3h=2a+a{\left(\frac{3k}{2a}\right)}^2$
$\Rightarrow 3h=2a+\frac{9k^2}{4a}$
$\Rightarrow 9k^2=4a(3h-2a)$
$\therefore$ Locus of centroid is
$y^2=\frac{4a}{3}(x-\frac{2a}{3})$
$\therefore$ Vertex $(\frac{2a}{3},0)$; directrix
$x-\frac{2a}{3}=-\frac{a}{3}\Rightarrow x=\frac{a}{3}$
and latus rectum $=\frac{4a}{3}$
$\therefore$ Focus $(\frac{a}{3}+\frac{2a}{3},0),i.e.(a,0)$.