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

Answer: 3

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