Question

In the given figure, two circles with centres O and P are touching internally at point A. If BQ = 9, DE = 5, complete the following activity to find the radii of the circles.

Answer 1

Let the radius of the bigger circle be R and that of a smaller circle be r.
OA, OB, OC and OD are the radii of the bigger circle.
∴ OA = OB = OC = OD = R
PQ = PA = r
OQ = OB − BQ = R - 9 

OE = OD − DE = R - 5

As the chords QA and EF of the circle with centre P intersect in the interior of the circle, so by the property of internal division of two chords of a circle,
OQ × OA = OE × OF
R - 9 x r = R - 5 x R - 5             .....(∵ OE = OF)
R² − 9R = R² − 10R + 25
R = 25 

AQ = 2r = AB − BQ                      (∵AB = 2R)
2r = 50 − 9 = 41
r = \[\frac{41}{2}\] = 20.5