As shown in figure three circles which have the same radius $r$ , have centres at $(0, 0), (1, 1), (2$ , 1). If they have a common tangent line, as shown, then the value of $105 r$ is.

Question

As shown in figure three circles which have the same radius r , have centres at ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 1). If they have a common tangent line, as shown, then the value of 10 5 ​ r is.

TestHub · Accepted Answer

Equation of line joining origin and centre of circle $C_{2} \equiv (2, 1)$ is, $y = \frac{x}{2}$

$\Rightarrow x - 2 y = 0$

Let equation of common tangent is $x - 2 y + c = 0$ .........(1)

∴ perpendicular distance from $(0, 0)$ on this line

= perpendicular distance from $(1, 1)$

$\Rightarrow \frac{c}{5} = \frac{c - 1}{5}$

$\Rightarrow c = 1 - c \Rightarrow c = \frac{1}{2}$

Equation of common tangent is

$x - 2 y + \frac{1}{2} = 0$ or $2 x - 4 y + 1 = 0$ ......(2)

Perpendicular from $(2, 1)$ on the line (2)

$r = \frac{4 - 4 + 1}{20} = \frac{1}{2 5} = \frac{5}{10}$

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