Question

Statement-1: The family of straight lines $2x{\sin }^2\theta +y{\cos }^2\theta =2\cos 2\theta (\theta \in R)$ passes through a fixed point $(a,b)$ where $a+b=1$.
Statement-2: $L_1+\lambda L_2=0(\lambda \in R)$ represents family of lines passing through point of intersection of two fixed lines $L_1=0$ and $L_2=0$.

• A. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement-2 is false.
• D. Statement-1 is false, statement-2 is true.

Answer 1

Answer: (A)

Statement-1: We have $2x{\sin }^2\theta +y{\cos }^2\theta =2\left({\cos }^2\theta -{\sin }^2\theta \right)$
$\Rightarrow 2x{\sin }^2\theta +2{\sin }^2\theta +y{\cos }^2\theta -2{\cos }^2\theta =0$
$\Rightarrow (2x+2){\sin }^2\theta +(y-2){\cos }^2\theta =0\Rightarrow (2x+2)+(y-2){\cot }^2\theta =0=L_1+\lambda L_2=0$
$\therefore \text{ The line passes through fixed point }(-1,2)\Rightarrow a=-1\text{ and }b=2\Rightarrow a+b=1$
So, S-1 is true. Obviosly S-2 is true and explaning S-1 also.