Question

Consider the region $R=\left\{(x, y) \in R \times R: x \geq 0\right.$ and $\left.y^{2} \leq 4-x\right\}$. Let $F$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $F$. Let $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y ^{2}=4- x$.
The value of $\alpha$ is ____

Answer 1

Let the equation of circle $(x-r)^{2}+y^{2}=r^{2}$
Equation of parabola $y^{2}=4-x$
Solving them
$(x-r)^{2}+4-x=r^{2} $
$\Rightarrow x^{2}-x(2 r+1)+4=0$
Since circle and parabola meet tangentially hence
$(2 r+1)^{2}-16=0 $
$\Rightarrow 2 r+1=4 $
$\Rightarrow r=3 / 2 $ and $ x^{2}-4 x+4=0$
$(x-2)^{2}=0 $
$\Rightarrow \alpha=2, r=3 / 2=1.5$