Question

Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of $\angle APB$ is
A. ${30^0}$
B. ${45^0}$
C. ${60^0}$
D. ${90^0}$

Answer 1

Hint: Draw the diagrams as given in the problem and use geometrical theorems involving tangents to the circle and triangles to solve the problem.

Complete step-by-step answer:

Given the problem, two circles touch each other externally at P.

Let O1 and O2 be the two circles touching each other externally at point P.
Also, AB is a common tangent to the circles touching them at A and B.
We need to find the value of $\angle APB$.

It is a well-known theorem that lengths of tangents drawn from an external point to a circle are equal.

Hence tangent AD = DP for circle O1 and tangent DB = DP for circle O2.

Also, angles opposite to equal sides of a triangle are equal.

Therefore, in $\Delta ADP$,
$\angle DAP = \angle DPA{\text{ (1)}}$
Similarly, in $\Delta DPB$
$\angle DPB = \angle DBP{\text{ (2)}}$

We know that the sum of all the angles within a triangle is equal to ${180^0}$.

Using this rule in $\Delta APB$ , we get,
$\angle PAB + \angle PBA + \angle BPA = {180^0}\,{\text{ (3)}}$

Also, from figure, it is clear that
$
  \angle BPA = \angle DPA + \angle DPB \\
  \angle PAB = \angle DAP \\
  \angle PBA = \angle DBP \\
$

Using the above obtained results in equation $(3)$ , we get,
$\angle DAP + \angle DBP + \angle DPA + \angle DPB = {180^0}$

Now using the result obtained in equations $(1)$and $(2)$ in above
\[
   \Rightarrow \angle DPA + \angle DPB + \angle DPA + \angle DPB = {180^0} \\
   \Rightarrow 2\left( {\angle DPA + \angle DPB} \right) = {180^0} \\
   \Rightarrow \left( {\angle DPA + \angle DPB} \right) = {90^0}{\text{ (4)}} \\
\]

As obtained earlier from figure, $\angle BPA = \angle DPA + \angle DPB$
Using this in equation $(4)$, we get
$\angle BPA = \angle APB = {90^0}$

Hence the value of $\angle APB$ is ${90^0}$.

Therefore, option (D), ${90^0}$ is the correct answer.

Note: The geometric theorems related to circles and triangles should be kept in mind in problems like above. Special attention should be paid while making the diagram from the problem as it should accurately depict the conditions given in the problem.