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  • If the point  (2cosθ, 2sinθ) does not lie in the  angle  between the lines x + y =2  and  x - y = 2 in which  the origin lies, t

If the point  (2cosθ, 2sinθ) does not lie in the  angle  between the lines x + y =2  and  x - y = 2 in which  the origin lies, then number of solutions of the  equation\sqrt{2} + cosθ +  sinθ = 0  is 

 

Option: 1

 0


Option: 2

 1


Option: 3

 2


Option: 4

3


Answers (1)

The  point  (2 cosθ, 2 sinθ)  lies  on the  circle \mathrm{x^2+y^2=4}. From the figure, it  is  obvious that \mathrm{\theta \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]}

\mathrm{\text { Now, } \cos \theta+\sin \theta=-\sqrt{2}}

\mathrm{\begin{aligned} & \Rightarrow-\sqrt{2} \cos \left(\theta-\frac{\pi}{4}\right)=\sqrt{ } 2 \\ & \Rightarrow \cos \left(\theta-\frac{\pi}{4}\right)=-1 \mid \\ & \text { Now, } \quad-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \\ & -\frac{3 \pi}{4} \leq\left(\theta-\frac{\pi}{4}\right) \leq \frac{\pi}{4} \\ & -\frac{1}{\sqrt{2}} \leq \cos \left(\theta-\frac{\pi}{4}\right) \leq 1 \end{aligned}}

Hence no solution. 

 

 

Posted by

Kshitij

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