Q. 495.0( 5 Votes )

# In the given figure, O is the center of two concentric circles of radii 5 cm and 3 cm. From an external point P tangents PA and PB are drawn to these circles. If PA = 12 cm then PB is equal to

A. 5 cm

B. 3√5 cm

C. 4√10 cm

D. 5√10 cm

Answer :

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 = 12 cm and OA = radius of outer circle = 5 cm

(OP)^{2} = (5)^{2} + (12)^{2}

(OP)^{2} = 25 + 144 = 136

OP = 13 cm …[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 = 3 cm

And, from [2] (OP) = 13 cm

(13)^{2} = (3)^{2} + (PB)^{2}

(PB)^{2} = 169 - 9 = 160

PB = 4√10 cm

Rate this question :

In Fig. 10.54, a circle touches all the four sides of a quadrilateral ABCD with AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

RD Sharma - Mathematics

If is isosceles with AB = AC and C (O, r) is the incircle of the touching BC at L, prove that L bisect BC.

RD Sharma - MathematicsIn Fig. 10.60, PA and PB are tangents from an external point P to a circle with centre O. LN touches the circle at A. Prove that PL + LM = PN + MN.

RD Sharma - Mathematics

In Fig 10.57, a circle is inscribed in a quadrilateral ABCD in which . If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius r of the circle.

RD Sharma - Mathematics

In fig. 10.55, O is the centre of the circle and BCD is tangent to it at C. Prove that .

RD Sharma - Mathematics

In Fig. 10.58, there are two concentric circles with centre O of radii 5 cm and 3 cm. from an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.

RD Sharma - Mathematics

Two tangent segments PA and PB are drawn to a circle with centre O such that . Prove that OP = 2 AP.

RD Sharma - Mathematics