Question

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

• A. $\frac{1}{2\sqrt{5}}$ units
• B. $5$ units
• C. $\frac{1}{\sqrt{5}}$ units
• D. $\sqrt{5}$ units

Answer 1

The correct option is A $\frac{1}{2\sqrt{5}}$ units


Given centres are $(0,0),(1,1)$ and $(2,1)$
The tangent is parallel to the line joining $(0,0)$ and $(2,1)$
$m=\frac{1-0}{2-0}=\frac{1}{2}$
Assuming the tangent line as
$x-2y+c=0\cdots \left(1\right)$
Internal point of simlitude lies on transverse common tangents
Hence it will pass through mid point $(1,1)$ and $(2,1)$ i.e., $(3/2,1)$ ($\because$ circles have equal radius)
putting the value in $\left(1\right)$ we get
$c=1/2$
Now using tangency condition
$r=\frac{1/2}{\sqrt{1+4}}=\frac{1}{2\sqrt{5}}$ units