In the given figure, AP and BP are tangents to a circle with centre O. If $$\angle$$APB = 62$$^\circ$$ then the measure of $$\angle$$AQB is:

Given, $$\angle$$APB = 62$$^\circ$$

AP and BP are tangents to the circle with centre O

$$\Rightarrow$$  $$\angle$$OAP = 90$$^\circ$$ and $$\angle$$OBP = 90$$^\circ$$

In quadrilateral OAPB,

$$\angle$$AOB + $$\angle$$OBP + $$\angle$$APB + $$\angle$$OAP = 360$$^\circ$$

$$\Rightarrow$$  $$\angle$$AOB + 90$$^\circ$$ + 62$$^\circ$$ + 90$$^\circ$$ = 360$$^\circ$$

$$\Rightarrow$$  $$\angle$$AOB + 242$$^\circ$$ = 360$$^\circ$$

$$\Rightarrow$$  $$\angle$$AOB = 118$$^\circ$$

Angle subtended by major arc at the centre is double the angle subtended by the major at any point on the circle.

$$\Rightarrow$$  $$\angle$$AOB = 2$$\angle$$AQB

$$\Rightarrow$$  118$$^\circ$$ = 2$$\angle$$AQB

$$\Rightarrow$$  $$\angle$$AQB = 59$$^\circ$$

Hence, the correct answer is Option D