Question

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

Answer 1

Given, O is the center 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.
Now, tangent drawn from an external point is perpendicular to the radius at the point of contact.
∴∠OAP = ∠OBP = 90°
∴ From right – angled ΔOAP,OP²  = OA² +  PA²

`⇒OP = sqrt(OA^2 + PA^2)`

`⇒ OP = sqrt (6^2 +10^2)`

`⇒ OP = sqrt( 136cm)`.

∴ From right – angled `Δ OAP,OP^2 = OB^2 + PB^2`

`⇒ PB = sqrt(OP^2 -OB^2)`

`⇒ PB = sqrt(136-16)`

`⇒ PB= sqrt(120cm)`

⇒ PB = 109.cm.

∴ The length of PB is 10.9 cm.