Question

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

• A. 8 only
• B. 2 only
• C. $\frac{1}{4}$ only
• D. any a $>0$

Answer 1

Answer: (D)

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