Sum

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.

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#### Solution

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)

R^{2} − 9R = R^{2} − 10R + 25

R = 25

AQ = 2r = AB − BQ (∵AB = 2R)

2r = 50 − 9 = 41

r = \[\frac{41}{2}\] = **20.5 **

Concept: Surface Area of a Combination of Solids

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