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

Answer 1

Answer: (A), (D)

focus is $(a,0)$
vertex is $\left(\frac{2a}{3},0\right)$

Let $P=(a{t}^2,2at)$
Therefore equation of tangent and normal at point P of the parabola are,

$ty=x+a{t}^2$ and $x+ty=2at+a{t}^2$ respectively
$\Rightarrow T=(-a{t}^2,0),N=(2at+a{t}^2,0)$

Let centroid of $\Delta PTN$ is $G\equiv (h,k)$

$\Rightarrow h=\frac{2a+a{t}^2}{3},k=\frac{2at}{3}$

$\Rightarrow \left(\frac{3h-2a}{a}\right)=\frac{9k^2}{4a^2}$

$\Rightarrow$ required parabola is

$\Rightarrow$$\frac{9y^2}{4a^2}=\frac{(3x-2a)}{a}=\frac{3}{a}(x-\frac{2a}{3})$

$\Rightarrow y^2=\frac{4a}{3}(x-\frac{2a}{3})$

Vertex $\equiv \left(\frac{2a}{3},0\right)$ , Focus $\equiv (a,0)$