Question

If $\theta$ is the acute angle between the tangents drawn from $(1,4)$ to the parabola $y^2=12x,$ then the value of $\sin \theta +2\cos \theta$ is equal to

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

Answer 1

The correct option is A $\sqrt{5}$

Equation of tangent is $y=mx+\frac{3}{m}$
Tangent passes through $(1,4).$
$\Rightarrow 4=m+\frac{3}{m}$
$\Rightarrow m^2-4m+3=0$
$\Rightarrow m=1,3$
$\therefore$ Slope of tangents are $m_1=3,m_2=1$
$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\frac{1}{2}$
$\Rightarrow \sin \theta =\frac{1}{\sqrt{5}}$ and $\cos \theta =\frac{2}{\sqrt{5}}$
$\therefore \sin \theta +2\cos \theta =\frac{1}{\sqrt{5}}+\frac{4}{\sqrt{5}}=\sqrt{5}$