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#### Chapters

1 - Real Numbers2 - Polynomials

3 - Pair of Linear Equations in Two Variables

4 - Quadratic Equations

5 - Arithmetic Progressions

6 - Triangles

7 - Coordinate Geometry

8 - Introduction to Trigonometry

9 - Some Applications of Trigonometry

10 - Circles

11 - Constructions

12 - Areas Related to Circles

13 - Surface Areas and Volumes

14 - Statistics

15 - Probability

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### Chapter 10 - Circles

#### Page 209

How many tangents can a circle have?

Fill in the blanks:

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

Fill in the blanks:

A line intersecting a circle in two points is called a __________.

Fill in the blanks:

A circle can have __________ parallel tangents at the most.

Fill in the blanks:

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) `sqrt119` cm test

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.

#### Pages 213 - 214

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 the given figure, 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 an 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 4 cm. 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

In Fig.2, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB + CD = BC + DA.

In the given figure, 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 segments joining the pointsof contact to the centre.

Prove that the paralleogram circumscribing a circle, is a rhombus

Prove that a 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 given figure). 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.

#### Extra questions

In fig., O is the centre of the circle, PA and PB are tangent segments. Show that the quadrilateral AOBP is cyclic.

From a point P, 10 cm away from the centre of a circle, a tangent PT of length 8 cm is drawn. Find the radius of the circle.

In the given figure, PQ is a chord of length 8cm of a circle of radius 5cm. The tangents at P and Q intersect at a point T. Find the length TP

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.

Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

In the given figure, the incircle of ∆ABC touches the sides BC, CA and AB at D, E, F respectively. Prove that AF + BD + CE = AE + CD + BF = `\frac { 1 }{ 2 } ("perimeter of ∆ABC")`

A circle touches the side BC of a ∆ABC at P, and touches AB and AC produced at Q and R respectively, as shown in the figure. Show that AQ = `\frac { 1 }{ 2 } `

Prove that there is one and only one tangent at any point on the circumference of a circle.

A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that AB + CD = AD + BC

In the given figure, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects PQ at A and RS at B. Prove that ∠AOB = 90º

In Fig., if AB = AC, prove that BE = EC

ABCD is a quadrilateral such that ∠D = 90°. A circle (O, r) touches the sides AB, BC, CD and DA at P,Q,R and If BC = 38 cm, CD = 25 cm and BP = 27 cm, find r.

Prove that the tangents at the extremities of any chord make equal angles with the chord.

In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length.

A point P is 13 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 12cm. Find the radius of the circle.

Prove that in two concentric circles, the chord of the larger circle which touches the smaller circle, is bisected at the point of contact.

In fig. XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that, XA + AR = XB + BR.

Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25 cm. Given that the radius of the circle is 7 cm.

In fig., circles C(O, r) and C(O’, r/2) touch internally at a point A and AB is a chord of the circle C (O, r) intersecting C(O’, r/2) at C, Prove that AC = CB.

PA and PB are tangents from P to the circle with centre O. At point M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.

ABC is a right triangle, right angled at B. A circle is inscribed in it. The lengths of the two sides containing the right angle are 6 cm and 8 cm. Find the radius of the incircle.

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