Question

If $\theta$ is the angle between the two tangents to $y^2=12x$ drawn from the point $(1,4)$, then $\tan \theta$ is equal to

• A. $\frac{2}{3}$
• B. $\frac{3}{4}$
• C. $\frac{3}{5}$
• D. $\frac{1}{2}$

Answer 1

The correct option is D $\frac{1}{2}$

Let equation of tangent to $y^2=4ax$ be
$y=mx+\frac{a}{m}$
Here, $a=3$
Since, it passes through $(1,4)$
$4=m+\frac{3}{m}\Rightarrow m^2-4m+3=0\therefore m_1=3,m_2=1$
Now,
$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|\Rightarrow \tan \theta =\left|\frac{3-1}{1+1\cdot 3}\right|\therefore \tan \theta =\frac{1}{2}$