Question

Let the equation $x+\lambda y-2 \sin \theta+\lambda(\cos \theta-1)=0$ represents a family of non-parallel lines for $a \theta\left(\right.$ where $\left.\theta \in\left[0, \frac{\pi}{2}\right]\right)$ 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

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