In the given figure, O is the center of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circle respectively. If PA = 10 cm, find the length of PB up to one place of decimal.
In given Figure,
OA ⏊ AP
[Tangent at any point on the circle is perpendicular to the radius through point of contact]
∴ In right - angled △OAP,
By Pythagoras Theorem
[i.e. (hypotenuse)2 = (perpendicular)2 + (base)2]
(OP)2 = (OA)2 + (PA)2
Given, PA = 10 cm and OA = radius of outer circle = 6 cm
(OP)2 = (6)2 + (100)2
(OP)2 = 36 + 100 = 136 [1]
Also,
OB ⏊ BP
[Tangent at any point on the circle is perpendicular to the radius through point of contact]
∴ In right - angled △OBP,
By Pythagoras Theorem
[i.e. (hypotenuse)2 = (perpendicular)2 + (base)2]
(OP)2 = (OB)2 + (PB)2
Now, OB = radius of inner circle = 4 cm
And from [2]
(OP)2 = 136
136 = (4)2 + (PB)2
(PB)2 = 136 - 16 = 120
PB = 10.9 cm