Question

The diameter of the circle, whose centre lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2,$ is_____.

Answer 1

$\because$ center lies on $x+y=2$ and in 1 st quadrant center $=(\alpha, 2-\alpha)$
where $\alpha>0$ and $2-\alpha >0$
$\Rightarrow 0<\alpha<2$
$\because \quad$ circle touches $x=3$ and $y=2$
$\Rightarrow \quad|3-\alpha|=|2-(2-\alpha)|=$ radius
$\Rightarrow |3-\alpha|=|\alpha|$
$\Rightarrow \alpha=\frac{3}{2}$
$\therefore \quad$ radius $=\alpha$
$\Rightarrow $ Diameter $=2 \alpha=3$