1. Tardigrade
  2. Question
  3. Mathematics
  4. Statement-1: The family of straight lines 2 x sin 2 θ+y cos 2 θ=2 cos 2 θ(θ ∈ R) passes through a fixed point (a, b) where a+b=1. Statement-2: L1+λ L2=0(λ ∈ R) represents family of lines passing through point of intersection of two fixed lines L 1=0 and L 2=0.

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Statement-1: The family of straight lines $2 x \sin ^2 \theta+y \cos ^2 \theta=2 \cos 2 \theta(\theta \in R)$ 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 $L _2=0$.

Straight Lines

Solution:

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