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  4. Let P: y 2=4 a x , a>0 be a parabola with focus S. Let the tangents to the parabola P make an angle of (π/4) with the line y=3 x+5 touch the parabola P at A and B. Then the value of a for which A , B and S are collinear is:

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Q. Let $P : y ^{2}=4 a x , a>0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ make an angle of $\frac{\pi}{4}$ with the line $y=3 x+5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A , B$ and $S$ are collinear is:

JEE MainJEE Main 2022Conic Sections

Solution:

Lines making angle $\frac{\pi}{4}$ with $y=3 x+5$ have slope $-2 \& 1 / 2$.
Which are perpendicular to each-other so, $A, S, B$ are collinear for all $a >0$.
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