Question

Find the angle between tangents at points of intersection of parabolas $y^2=4ax$ and $x^2=4ay$

• A. $\frac{\pi }{2}$
• B. $\arctan \frac{3}{4}$
• C. $\frac{\pi }{4}$
• D. $\arctan \frac{-3}{4}$

Answer 1

Answer: (A), (B)

The correct options are
A $\frac{\pi }{2}$
B $\arctan \frac{3}{4}$

Point of intersections are $(0,0)$ and $(4a,4a)$
equation of tangents at $(0,0)$ are $0.y=2a(x+0)$ $\Rightarrow x=0$
and $0.x=2a(y+0)$ $\Rightarrow y=0$
Therefore, angle between tangents is $\frac{\pi }{2}$
equation of tangents at $(4a,4a)$ are $4a.y=2a(x+4a)$ $\Rightarrow x-2y+4a=0$
and $4a.x=2a(y+4a)$ $\Rightarrow y-2x+4a=0$
Therefore, angle between tangents is ${\tan }^{-1}\left|\frac{2-\frac{1}{2}}{1+1}\right|={\tan }^{-1}\frac{3}{4}$