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Length of Latus Rectum

The tangent P...

Question

The tangent PT and the normal PN 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 PTN is a parabola, whose

vertex is

(\frac{2 a}{3}, 0)

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directrix is x = 0

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latus rectum is

\frac{2 a}{3}

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focus is (a, 0)

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Solution

The correct options are
A vertex is $(\frac{2 a}{3}, 0)$
D focus is (a, 0)
Equation of tangent and normal at point $P (a t^{2}, 2 a t)$ is
$t y = x + a t^{2}$ and $y = - t x + 2 a t + a t^{3}$
Let centroid of $Δ P T N$ is $R (h, k)$ .
$∴ h = \frac{a t^{2} + (- a t^{2}) + 2 a + a t^{2}}{3}$

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

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