Question

Angle between the tangents to the parabola $y^2=8x$ from the point $(6,7)$ is

• A. ${\tan }^{-1}\left(\frac{1}{4}\right)$
• B. ${\tan }^{-1}\left(1\right)$
• C. ${\tan }^{-1}\left(\frac{1}{2}\right)$
• D. None of these

Answer 1

Answer: (D)

None of these

The equation of tangent to the parabola
$y^2=8x$ is $y=mx+\frac{2}{m}$
This tangent passes through the point $(6,7)$
$\therefore 7=6m+\frac{2}{m}$
$\Rightarrow 6m^2-7m+2=0$
$\Rightarrow (3m-2)(2m-1)=0$
$\Rightarrow m=\frac{2}{3},\frac{1}{2}$
$\therefore$ Required angle, $\theta ={\tan }^{-1}\left(\frac{\frac{2}{3}-\frac{1}{2}}{1+\frac{2}{3}\times \frac{1}{2}}\right)$
$={\tan }^{-1}\left(\frac{\frac{4-3}{6}}{\frac{4}{3}}\right)={\tan }^{-1}\left(\frac{1}{8}\right)$