Question

Let the equation $x+\lambda y-2\sin \theta +\lambda (\cos \theta -1)=0$ represents a family of non-parallel lines for $a\theta ($ where $\theta \in \left[0,\frac{\pi }{2}\right])$ which passing through a fixed point $(p,q)$ and distance of a point $(0,1)$ from the point $(p,q)$ is the greatest, then find the value of $(p+q)$.

Answer 1

Answer: 3

$(x-2\sin \theta )+\lambda (y+\cos \theta -1)=0$
$x=2\sin \theta ,y=1-\cos \theta$
$\therefore (p,q)\equiv (2\sin \theta ,1-\cos \theta )$
$\text{ Now, }D=\sqrt{(2\sin \theta -0{)}^2+(1-\cos \theta -1{)}^2}$
$D=\sqrt{4{\sin }^2\theta +{\cos }^2\theta }$
${\therefore D|}_{\max }=\sqrt{3{\sin }^2\theta +1}=2\text{ when }\theta =\frac{\pi }{2}$
$\therefore (p,q)\equiv (2,1)\Rightarrow p+q=3$