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Variable Separable Method

The tangent ...

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 $△ P T N$ is a parabola whose

vertex =

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

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directrix is

x = 0

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latus rectum =

(\frac{2 a}{3})

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focus

= (a, 0)

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Solution

The correct options are
A vertex = $(\frac{2 a}{3}, 0)$
D focus $= (a, 0)$
Equation of tangent and normal at point $P (a t^{2}, 2 a t)$ is $t y = x + {a t}^{2}$ and $y = - t x + 2 a t + a t^{2}$
Let centroid of $△ P T N$ is $R (h, k)$
$∴ h = \frac{{a t}^{2} + ({- a t}^{2}) + 2 a + {a t}}{3}$
and $k = \frac{2 a t}{3} \Rightarrow 3 h = 2 a + a . {(\frac{3 k}{2 a})}^{2}$
$\Rightarrow 3 h = 2 a + \frac{{9 k}^{2}}{4 a} \Rightarrow {9 k}^{2} = 4 a (3 h - 2 a)$
Therefore Locus of centroid is $y^{2} = \frac{4 a}{3} (x - \frac{2 a}{3})$
And vertex $(\frac{2 a}{3}, 0)$ ; Directrix $x - \frac{2 a}{3} = - \frac{a}{3} \Rightarrow x = \frac{a}{3}$
Latus rectum $= \frac{4 a}{3}$
Therefore Focus $(\frac{a}{3} + \frac{2 a}{3}, 0) \Rightarrow (a, 0)$

Suggest Corrections

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