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

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____________

Solution:

$\because \quad$ 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

$\Rightarrow|3-\alpha|=|\alpha| \Rightarrow \alpha=\frac{3}{2}$

$\therefore \quad$ radius $=\alpha$

$\Rightarrow$ Diameter $=2 \alpha=3$

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