Straight Line And Pair Of Straight Lines Ques 13
- The equation $y=\sin x \sin (x+2)-\sin ^{2}(x+1)$ represents a straight line lying in
(a) second and third quadrants only
(b) first, second and fourth quadrants
(c) first, third and fourth quadrants
(d) third and fourth quadrants only
Show Answer
Solution:
Formula:
Equation Of A Straight Line In Various Forms:
Key Idea Use formulae :
$2 \sin A \sin B=\cos (A-B)-\cos (A+B)$ and $\cos 2 \theta=1-2 \sin ^{2} \theta$
Given equation is $y=\sin x \sin (x+2)-\sin ^{2}(x+1)$
$$ =\frac{1}{2}[\cos 2-\cos (2 x+2)]-\frac{1}{2}[1-\cos (2 x+2)] $$
$[\because 2 \sin A \sin B=\cos (A-B)-\cos (A+B)$ and $\left.\cos 2 \theta=1-2 \sin ^{2} \theta \Rightarrow 2 \sin ^{2} \theta=1-\cos 2 \theta\right]$
$=\frac{1}{2} \cos 2-\frac{1}{2} \cos (2 x+2)-\frac{1}{2}+\frac{1}{2} \cos (2 x+2)$
$=\frac{1}{2}(\cos (2)-1)=-\frac{1}{2}\left(2 \sin ^{2}(1)\right)$
$=-\sin ^{2}(1)<0 \Rightarrow y<0$
and as we know that $y<0$, is in third and fourth quadrants only.
BETA
