Question

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60° then find the length of OP.

Answer 1

Let PA and PB be the two tangents drawn to the circle with centre O and radius a such that $\angle \mathrm{APB}=60°.$



In ∆OPB and ∆OPA
OB = OA = a (Radii of the circle)
$\angle \mathrm{OBP}=\angle \mathrm{OAP}=90°$ (Tangents are perpendicular to radius at the point of contact)
BP = PA (Lengths of tangents drawn from an external point to the circle are equal)
So, ∆OPB ≌ ∆OPA (SAS Congruence Axiom)
$\therefore \angle \mathrm{OPB}=\angle \mathrm{OPA}=30°$ (CPCT)
Now,
In ∆OPB,
$$\begin{gathered} \sin 30°=\frac{\mathrm{OB}}{\mathrm{OP}} \\ \Rightarrow \frac{1}{2}=\frac{a}{\mathrm{OP}} \\ \Rightarrow \mathrm{OP}=2a \end{gathered}$$
Thus, the length of OP is 2a.
Disclaimer: The answer given in the book is incorrect.