Question

If a tangent to the parabola $y^2=8x$ makes an angle $\frac{\pi }{4}$ with the straight line $y=3x+5$, then its point of contact may be

• A. $(\frac{1}{2},2)$
• B. $(8,8)$
• C. $(\frac{1}{2},-2)$
• D. $(-\frac{1}{2},2)$

Answer 1

Answer: (B), (C)

The correct options are
B $(8,8)$
C $(\frac{1}{2},-2)$

Let the point of contact be $(2t^2,4t)$, then slope of tangent is $\frac{1}{t}$
$\therefore \tan \frac{\pi }{4}=\left|\frac{\frac{1}{t}-3}{1+\frac{3}{t}}\right|=\left|\frac{1-3t}{t+3}\right|$
On solving we get,
$t=2,-\frac{1}{2}$
$\therefore$ points of contact are $(8,8)$, and $(\frac{1}{2},-2)$
Hence, option 'B' and 'C' are correct.