Question

Let $\left(a , \beta \right)$ be an ordered pair of real numbers satisfying the equation $x^{2}-4x+y^{2}+3=0.$ If the maximum and minimum values of $\sqrt{a^{2} + \beta ^{2}}$ are $l$ and $s$ respectively, then the value of $2\left(\frac{l - s}{l + s}\right)$ is equal to

Answer 1

$\sqrt{a^{2} + \beta ^{2}}$ represents the distance of $\left(a , \beta \right)$ from the origin $\left(0,0\right)$ .
Now, let the line joining $O\left(0 , 0\right)$ and center of circle $C\left(2,0\right)$ cuts the circle at points $\left(1,0\right)=A$ and $\left(3,0\right)=B$ ,then, $l=OB=3$ and $s=OA=1$
$\Rightarrow 2\left(\frac{l - s}{l + s}\right)=1$