two circles with center A and B touch each other externally. PM=15cm is tangent to circle with centre A and QN=13cm is tangent to circle with centre B from external points Pand Q. if PA=17cm and BQ=12cm. find distance between centre A and B of circle

The angle between a tangent and radius is right angle.angles BNQ and AMP are right angles
The triangle PAM is a right angled triangle, with PA being the hypotenuse.

Using Pythagoras theorem, the radius of the circle with centre A is found as
AM = Sqrt ( 17² - 15²)
AM = 8 cm

Similarly, the radius of circle with centre B is found as

BM= sqrt( 13² -12²)
BM = 5cm

Clearly , the distance between the centre is the sum of radii of the two circles ( AC +BC)

Required distance = 5cm + 8cm = 13 cm