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Circles Ex 10.2 Solutions

Question - 1 : -

From a point Q, thelength of the tangent to a circle is 24 cm and the distance of Q from thecentre is 25 cm.

Answer - 1 : -

 The radius of the circle is

(A) 7 cm                  (B) 12 cm

(C) 15 cm                (D) 24.5 cm

Answer:

First, draw a perpendicular from the center Oof the triangle to a point P on the circle which is touching the tangent. Thisline will be perpendicular to the tangent of the circle.

So, OP is perpendicular to PQ i.e. OP ⊥ PQ

From the above figure, it is also seen that △OPQ is a right angled triangle.

It is given that

OQ = 25 cm and PQ = 24 cm

By using Pythagoras theorem in △OPQ,

OQ2 = OP2 +PQ2

(25)2 = OP2+(24)2

OP2 = 625-576

OP2 = 49

OP = 7 cm

So, option A i.e. 7 cm is the radius of thegiven circle.

Question - 2 : -

In Fig. 10.11, if TPand TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, 

Answer - 2 : -

then ∠PTQ is equal to

(A) 60°                      (B) 70°

(C) 80°                      (D) 90°

Answer:

From the question, it is clear that OP is theradius of the circle to the tangent PT and OQ is the radius to the tangents TQ.

So, OP ⊥ PT and TQ ⊥ OQ

∴∠OPT = ∠OQT = 90°

Now, in the quadrilateral POQT, we know thatthe sum of the interior angles is 360°

So, ∠PTQ+∠POQ+∠OPT+∠OQT = 360°

Now, by putting the respective values we get,

∠PTQ +90°+110°+90° = 360°

∠PTQ = 70°

So, ∠PTQ is 70° which isoption B.

Question - 3 : - If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°.

Answer - 3 : -

then ∠ POA is equal to

(A) 50°                            (B) 60°

(C) 70°                            (D) 80°

Solution

Question - 4 : -

Prove that thetangents drawn at the ends of a diameter of a circle are parallel.

Answer - 4 : -

Question - 5 : -

Prove that theperpendicular at the point of contact to the tangent to a circle passes throughthe center.

Answer - 5 : -

Question - 6 : -

The length of atangent from a point A at distance 5 cm from the centre of the circle is 4 cm.Find the radius of the circle.

Answer - 6 : -

Question - 7 : -

Two concentriccircles are of radii 5 cm and 3 cm. Find the length of the chord of the largercircle which touches the smaller circle.

Answer - 7 : -

Question - 8 : -

A quadrilateral ABCDis drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD +BC

Answer - 8 : -

Question - 9 : - In Fig. 10.13, XY and X′Y′ are two paralleltangents to a circle with centre O and another tangent AB with point of contactC intersecting XY at A and X′Y′ at B. Prove that ∠AOB = 90°.

Answer - 9 : -

Question - 10 : -

Prove that the anglebetween the two tangents drawn from an external point to a circle issupplementary to the angle subtended by the line-segment joining the points ofcontact at the center.

Answer - 10 : -

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