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

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