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  4. Let P be a point on the parabola y2 = 4ax with focus F. Let Q denote the foot of the perpendicular from P onto the directrix. Then (tan ∠PQF/tan ∠PFQ) is

Q. Let P be a point on the parabola $y^2 = 4ax$ with focus F. Let Q denote the foot of the perpendicular from P onto the directrix. Then $\frac{tan \,∠PQF}{tan \,∠PFQ}$ is

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Solution:

Equation of parabola is
$y^{2}=4 a x ..... (i) $
image
Let the parametric coordinate of point $P$ on the parabola is
$(a, 2 a)$. Now, $Q F=2 \sqrt{2} a$
$P Q=2 a$ and $P F=2 a$
we observe that, $Q F^{2}=P Q^{2}+P F^{2}$
$\Rightarrow 8 a^{2}=4 a^{2}+4 a^{2}=8 a^{2}$
So, $\Delta Q P F$ form a right angle isoceles triangle.
In which, $\angle P Q F=\angle P F Q$
$\Rightarrow \tan \angle P Q F=\tan \angle P F Q$
$\Rightarrow \frac{\tan \angle P Q F}{\tan \angle P F Q}=1$

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