In the given figure, O is the centre of each one of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to outer and inner circle respectively. If PA = 10 cm, find the length of PB, upto two places of decimal.

Given,

O is the centre of two concentric circles of radii OA = 6 cm and OB = 4 cm.

PA and PB are the two tangents to the outer and inner circles respectively and PA = 10 cm.

We know that,

The tangent at any point of a circle and the radius through this point are perpendicular to each other.

∠OAP = ∠OBP = 90°

Since, OAP is a right angled triangle,

∴ OP² = OA² + PA² [By pythagoras theorem]

⇒ OP² = 6² + 10²

⇒ OP² = 36 + 100

⇒ OP² = 136

⇒ OP = $\sqrt{136}$

Since, OBP is a right angled triangle,

⇒ OP² = OB² + PB²

⇒ PB² = OP² - OB²

⇒ PB² = $(\sqrt{136})^2 - 4^2$

⇒ PB² = 136 - 16

⇒ PB² = 120

⇒ PB = $\sqrt{120}$

⇒ PB = 10.95 cm

Hence, length of PB is 10.95 cm.