Question

Find the length of the tangent segment AB to the circles centered at O and O' whose radii are a and b respectively when the circles touch each other

Answer 1

`AB=2√(ab)`

Explanation:

From figure,

`OA=a, \quad \quad O'B=b\\\Rightarrow OO'= (a+b) \quad \quad \text{and}\quad OD=(a-b)`

In triangle
`OO'D`

`(OO')^2=(OD)^2+(O' D)^2`

`\Rightarrow (a+b)^2=(a-b)^2+(O' D)^2\\\Rightarrow a^2+b^2+2ab-a^2-b^2+2ab=(O' D)^2\\\Rightarrow 4ab=(O' D)^2\\\Rightarrow O'D=2√(ab) \\\Rightarrow O' D=2√(ab)=AB \quad \quad [\because O' DAB\;\; \text{is a rectangle.}]`

Hence,
`AB=2√(ab)`