Question

Consider a quadratic polynomial ' $C^{\prime }$ and a line ' $l$ ' as $C:y=x^2-2x\cos \theta +\cos 2\theta +\cos \theta +\frac{1}{2}$ where $\theta \in \left[0,360^{\circ }\right]$ and $l:y=x$.
If $C$ touches the line $l$ then sum of all possible values of $\theta$, is

• A. $2\pi$
• B. $3\pi$
• C. $4\pi$
• D. $8\pi$

Answer 1

Answer: (C)

If $C$ touches $l$ then discriminant $=0$
$\Rightarrow x^2-(2\cos \theta +1)x+2{\cos }^2\theta +\cos \theta -\frac{1}{2}=0$ has equal roots
So, $(2\cos \theta +1{)}^2=4\left(2{\cos }^2\theta +\cos \theta -\frac{1}{2}\right)$
$\Rightarrow 4{\cos }^2\theta +4\cos \theta +1=8{\cos }^2\theta +4\cos \theta -2$
$\Rightarrow 4{\cos }^2\theta =3\Rightarrow {\cos }^2\theta =\frac{3}{4}$
$\theta =\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6}\Rightarrow$ sum $=\frac{24\pi }{6}=4\pi$