Question

Let α1 be a circle with centre O and AB be diameter. Let P be a point on OB different from O. Suppose another circle α2 with centre P lies in the interior of α1. Tangents are drawn from A and B to the circle α2 intersecting α1 at A1 and B1 respectively such that A1 and B1 are on opposite side of AB. If A1B = 6, AB1 = 18 and OP = 10, find radius of α1

Answer 1

Step-by-step explanation:

Given Let α1 be a circle with centre O and AB be diameter. Let P be a point on OB different from O. Suppose another circle α2 with centre P lies in the interior of α1. Tangents are drawn from A and B to the circle α2 intersecting α1 at A1 and B1 respectively such that A1 and B1 are on opposite side of AB. If A1B = 6, AB1 = 18 and OP = 10, find radius of α1

• So there is a circle α1 with centre O and diameter AB. P is a point on OB a bit forward from O. Also another circle α2 with centre P is on the interior of α1.  
• Now A1 and B1 are on opposite side of AB.
•                  Let OP = b and AB = a
•       Now from the triangle APQ we have  
•                                     Sin theta = b/a + 10 ------1
•    From triangle AA1B we get
•                                      Sin theta = 6/2a ----------2
•            From 1 and 2 we get
•                              So b / a + 10 = 6/2a ------------3
•          Similarly we get
•                             So b / a – 10 = 18 / 2a -----------4
•                      6a + 60 = 18a – 180
•                      6a – 18 a = - 180 – 60
•                         Or – 12 a = - 240
•                         Or a = 240 / 12
•                           Or a = 20
• Therefore radius of α1 = 20

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