The tangent and the normal to the parabola at a point on it meet its axis at points and , respectively. The locus of the centroid of the triangle is a parabola whose [2009]

Question

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 [2009]

Smart Achievers · Accepted Answer

(1, 4)

Let $P (a t^{2}, 2 a t)$ be any point on the parabola $P$

$∴$ Tangent to the parabola at P is $y = \frac{x}{t} + a t,$

which meets the axis of parabola i.e. $x$ -axis at $T (- a t^{2}, 0) .$

Also normal to parabola at P is $t x + y = 2 a t + a t^{3}$

which meets the axis of parabola at $N (2 a + a t^{2}, 0) .$

Let $G (x, y)$ be the centroid of $∆ P T N$ , then

$x = \frac{a t^{2} - a t^{2} + 2 a + a t^{2}}{3}$ and $y = \frac{2 a t}{3}$

$\Rightarrow x = \frac{2 a + a t^{2}}{3} ...(i)$ and $y = \frac{2 a t}{3} ...(ii)$

Eliminating $t$ from (i) and (ii), we get the locus of centroid $G$ as

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

which is a parabola with vertex $(\frac{2 a}{3}, 0),$ 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 $(a, 0) .$

Solution

Q.
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 [2009]

1 vertex is $(\frac{2 a}{3}, 0)$

2 directrix is $x = 0$

3 latus rectum is $\frac{2 a}{3}$

4 focus is $(a, 0)$

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Solution

Q. 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 triangle PTN is a parabola whose [2009]

1 vertex is (2a3,0)

2 directrix is x=0

3 latus rectum is 2a3