The tangent $P T$ and the normal $P N$ to the parabola $y^2=4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $P T N$ is a parabola whose

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Aicharya · Accepted Answer

Let $P\left(a t^2, 2 a t\right)$ be any point on the parabola $y^2=4 a x$ $\therefore \quad$ Tangent to the parabola at $P$ is $y=\frac{x}{t}+a t$, which meets the axis of parabola i.e $x$-axis $a t$ $T\left(-a t^2, 0\right)$. Also normal to parabola at $P$ is $t x+y=2 a t+a t^3$ which meets the axis of parabola at $N\left(2 a+a t^2, 0\right)$ Let $G(x, y)$ be the centriod of $\triangle P T N$, then $x=\frac{a t^2-a t^2+2 a+a t^2}{3} \text { and } y=\frac{2 a t}{3}$ $\Rightarrow \quad x=\frac{2 a+a t^2}{3} \quad \ldots$ (i) and $y=\frac{2 a t}{3} \quad \ldots$ (ii) Eliminating $t$ from (i) and (ii), we get the locus of centriod $G$ as $3 x=2 a+a\left(\frac{3 y}{2 a}\right)^2 \Rightarrow y^2=\frac{4 a}{3}\left(x-\frac{2}{3} a\right)$, which is a parabola with vertex $\left(\frac{2 a}{3}, 0\right)$, directrix as $x-\frac{2 a}{3}=-\frac{a}{3} \Rightarrow x=\frac{a}{3}$, latus rectum as $\frac{4 a}{3}$ and focus as $(a, 0)$.

The tangent \(P T\) and the normal \(P N\) to the parabola \(y^2=4 a x\) at a point \(P\) on it meet its…

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