Two circles C1 and C2 touch each other internally at Q with centres O and O′ respectively. If radii of bigger and smaller circles are 4 cm and 3 cm respectively and ACDB is the straight line of 2 15 cm, then find the lengths of OA and AC.
Answers
Given : two circles C1 and c2 touch each other internally at Q with centres O and O' respectively. radii of bigger and smaller circles are 4 cm and 3 cm respectively and ACDB is the straight line of 2√15 cm
To Find : lengths of OA and AC
Solution :
AB = 2√15 cm
AB is chord of circle C2
Radius of C2 = 4 cm and center O'
O'A = O'B = 4cm
intersection of PQ & AB = M
AM = BM = AB/2 = √15 cm
O'A² = O'M² + AM²
=> 4² = O'M² + (√15)²
=> 16 = O'M² +15
=> O'M² = 1
=> O'M =1
O'Q = 4 cm
OQ = 3cm
=> OO' = 1 cm
O'M =1
=> OM = 1 + 1 = 2 cm
OA² = OM² + AM²
=> OA² = 2² + (√15)²
=> OA² = 4 + 15
=> OA = √19 cm
OC² = OM² + CM²
=> 3² = 2² + CM²
=> CM² = 5
=> CM = √5 cm
AC = AM - CM = √15 - √5
= √5(√3 - 1) cm
OA = √19 cm
AC = √5(√3 - 1) cm
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