Question

If two circles of radius $25$ units and $16$ units touch each other externally, then the radius of the circle which touches both of them externally and also their direct common tangent is

• A. $20\text{units}$
• B. $\frac{400}{81}\text{units}$
• C. $\frac{41}{2}\text{units}$
• D. $\frac{9}{2}\text{units}$

Answer 1

The correct option is B $\frac{400}{81}\text{units}$


We know that for two circle touching externally, length of direct common tangent
$AB=\sqrt{d^2-(r_1-r_2{)}^2}\Rightarrow AB=2\sqrt{r_1{r}_2}(\because d=r_1+r_2)\Rightarrow AB=2\sqrt{25\cdot 16}=40$

Let radius of required circle be $r$
We know that,
$AB=AP+PB\Rightarrow 40=2\sqrt{25\cdot r}+2\sqrt{r\cdot 16}\Rightarrow 40=18\sqrt{r}\therefore r={\left(\frac{20}{9}\right)}^2=\frac{400}{81}\text{units}$