Question

Consider the region $R=\{(x,y)\in R\times R:x\ge 0$ and $y^2\le 4-x\}$. 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

Answer: 2

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