Two circle with radii `r_(1)` and `r_(2)` respectively touch each other externally. Let `r_(3)` be the radius of a circle that touches these two circle as well as a common tangents to two circles then which of the following relation is true

Options

A. `(1)/(sqrt(a))-(1)/(sqrt(b))=(1)/(sqrt(c))`
B. `c=(2ab)/(a+b)`
C. `(1)/(sqrt(a))+(1)/(sqrt(b))=(1)/(sqrt(c))`
D. `c=(2ab)/(sqrt(a)+sqrt(b))`

Answer

3

Explanation

`:' M'N'=MN`
Now, `MN=sqrt((a+b)^(2)+(a-b)^(2))=sqrt(4ab)`
`N'O=sqrt((b+c)^(2)-(b-c)^(2))=sqrt(4bc)`
`M'O=sqrt((b+c)^(2)-(a-c)^(2))=sqrt(4ac)`
Now, `N'O +M'O=MN`
`implies 2sqrt(ab)=2sqrt(bc)+2sqrt(ac)`
Dividing by `(1)/(sqrt(abc))`, we get
`(1)/(sqrt(c ))=(1)/(sqrt(a))+(1)/(sqrt(b))`