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PN is an ordinate of the parabola $y^2=4 a x$ . A straight line is drawn through the middle point M of PN parallel to the axis meeting the parabola at Q . NQ meets the tangent at the vertex A, at a point T, then $AT / NP=$
Option: 1
0

Option: 2
2

Option: 3
$\frac{2}{3}$

Option: 4
5

Answers (1)

best_answer

Let the coordinates of P be $\left(a t^2, 2 a t\right)$ , then the coordinates of N are $\left(a t^2, 0\right)$ and of M are $\left(a t^2, a t\right)$
Equation of the line through $M\left(a t^2, a t\right)$ parallel to the axis is $y=a t$ which meets the parabola $y^2=4 a x$ at $Q\left(a t^2 / 4, a t\right)$

Equation of NQ is $\frac{y-0}{a t-0}=\frac{x-a t^2}{\frac{a t^2}{4}-a t^2}$

$\Rightarrow \quad\: \: \: \: \: \: \: \: \: y=-\frac{4}{3 t}\left(x-a t^2\right)$

which meets the tangents $x=0$ at the vertex A at the point T $\left(0, \frac{4 a t}{3}\right)$

So that, $A T=\frac{4 a t}{3}, \text { Also } NP=2 a t$

Hence, $=\frac{A T}{N P}=\frac{4 a t}{3 \times 2 a t}=\frac{2}{3}$

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HARSH KANKARIA

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