Question

Consider a quadratic polynomial ' $C^{\prime}$ and a line ' $l$ ' as $C : y = x ^2-2 x \cos \theta+\cos 2 \theta+\cos \theta+\frac{1}{2}$ 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 $=\frac{24 \pi}{6}=4 \pi$