Solution of Chapter 10. Circles (NCERT - Mathematics Book)

How many tangents can a circle have?

Fill in the blanks :

(i) A tangent to a circle intersects it in point (s).

(ii) A line intersecting a circle in two points is called a .

(iii) A circle can have parallel tangents at the most.

(iv) The common point of a tangent to a circle and the circle is called.

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :

A. 12 cm B. 13 cm

C. 8.5 cm D. 119 cm.

Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is

A. 7 cm B. 12 cm

C. 15 cm D. 24.5 cm

In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to

A. 60° B. 70°

C. 80° D. 90°

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to

A. 50° B. 60°

C. 70° D. 80°

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

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

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

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that

AB + CD = AD + BC

In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Prove that the parallelogram circumscribing a circle is a rhombus.

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.