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

Answer:

$AB=2\sqrt{ab}$

Step-by-step explanation:

From figure,

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

In triangle $OO'D$

  $(OO')^2=(OD)^2+(O' D)^2$

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

Hence, $AB=2\sqrt{ab}$