Circles 5 I 00~ BASIC CONCEPTS WITH EXAMPLES 4~ If a line segment whose end points lie on the circle is called c hord to the circle. 9 A101 ILIf a circle and a line have no common point, then line is called a non-intersecting line with respect to the circle. _PG Q kw If a circle and a line have two common points or a line intersect a circ le in two distinct points, then fine is called secant to the circle. 0. A P

4, If a line and a circle have only one point common, or a line intersect t he circle in only one point, then it is called tangent to the circle. _-0.0 V There is only one tangent at a point on the circumference of the circle. w, The common point of the tangent and the circle is called the poi nt of contact. 4~ The tangent at any point of a circle is perpendicular to the rad ius through the point of contact. %w The line containing the radius through the point of contact of tangent i s called the normal to the circle at the point: There is no tangent to the circl e passing through a point lying inside the circle. Q~ There are exactly two tangents to a circle through a point lying outside the circle. 4,' The length of the segment of the tangent from the external point and the point of contact with the circle is called the length of the tangent. t~ are equal. 1. T The lengths of tangents drawn from an external point to a circle

In figure, PT is a tangent and PAB is a secant. If PT = 6cm, AB = 5 cm, rind the length of PA. [Al 20061

P-'c:~ A B Sol. T - -------- 40 P Circles 85 .

-

Join OT, OA and OP. Draw OC 1 A113. Let radius of circle = r.

I LONG ANSWER TYPE QUESTIONS I PREVIOUS YEARS' QUESTIONS 1. ProNe that the lengths of tangents drawn from an external D C

point to a circle are equal.

Using the above, prove the following: A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC. A B

/D

[Delhi 2008, At 20091 Prove that the lengths of the tangents drawn from an exter nal point to a circle are equal. Using the above, do the following: In the fig., TP and TQ are tangents from T to the circle with centre 0 and R is any point on the circle. If AB is a tangent to the circle at R, prove that TA + AR = TB + BR . Cp A 0. >T B Q [At 20081 3. Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above, prove the following: ABC is an isosceles triangle in which AB = AC, circumscribed about a circle, as shown in the fig. Prove that the base is bisected by the point of contact.

A R B Q P C

[Foreign 2008] 4. Prove that the tangent drawn from an external point to a circle are equal. Using the above, do the following.

Two concentric circles are of radii 5 cm and 3 cm, find the length of the chord of the larger circle which is tangent to the smaller circle. [Delhi 2008C] 5. Prove that the lengths of tangents drawn from an external point to a cir cle are equal. Making use of the above, prove the following: From an external point Ptwo tangents PA and PB are drawn to a circle with centre 0 as shown in figure. Show that OP is the perpendicular bisector of AB. A [At 2008 C1 6. (a) Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following: In figure, 0 is the centre of the two concentric circles. 94 ?b9aAm a0e CCE (Mathematics - X) AB is a chord of the larger circle touching the smaller circle at C. Prove that AC = BC. 00 A 1~ ZMLO = 30', ZRLO = 90'. 14. A x 0..-.-- D f E B C Join OB and OD Now ZOBC = 90', ZDBC = 180' - 146' = 34' ZOBD = 90' - 340 * 56' Find ZBOD and use ZBOD = 2x. SHORT ANSWER TYPE QUES'nONS-11 2. Let AD BD CF CF Also Now BC = 8 Solve W and 00. 4. Join OC Now ZACB = 90' .*. ZABC = 60* .*. BD=12-x => BE=12-x CE = y AF=x X+Y= 10 12-x+y=8 =x =BE =.v =CE AD =AF AC = 10

...

W ...

(ii)

Also OB = OC =* ZOBC = ZOCB = 60 Now ZCBD = 180' - 60' = 120' and ZBCD = 90o - 6 0' = 30.*. ZBDC = 30' =:~ BC = BD A 0 B D D 9. Draw OD I BC Join OC and OT use Pythagoras Theorem. Circles 97 ~ime Allowed: 1 Hour TEST YOUR KNOWLEDGE-1 1. The length of a tangent from a point P at a distance 5 cm from t he centre of the circle is 4 cm. Find the radius of the circle. Maximum Marks: 251

2. Prove that, in two concentric circles, the chord of the larger circle wh ich touches the smaller circle, is bisected at the point of contact. 3. Find the length of tangent to a circle with radius 5 cm drawn fr om a point at a distance of 13 cm from the centre of the circle. 4. Prove that the tangents at the extremities of any chord make equ al angles with the chord. 5. Prove that the angle between the two tangents drawn from an external poi nt to a circle is supplementary to the angle subtended by the line segment joini ng the points of contact at the centre. 6. A circle touches all the four sides of a quadrilateral ABCD whos e side AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD. 7. The tangent at any point of a circle is perpendicular to the rad ius through point of contact. Prove it. Use the result to solve the following : From a point P, the length of tangent to a circle is 12 cm and the dista nce of P from the centre is 13 cm. Find the radius of the circle. 8. Number of tangents to a circle which are parallel to a secant is (a) zero (b) 2 (C) I (d) infinite 9. Two circles touch internally at a point P, tangent segments TQ, TR are drawn to the two circles, then (a) TQ = TR (b) PQ = TR (c) PQ = TQ (d) None of these 10. Two parallel lines touch the circle at points A and B separately . If area of the circle is 25 n cm', then AB is equal to (a) 8 cm (b) 5 cm (C) 10 cm (d) 25 cm TEST YOUR KNOWLEDGE-2 1. Find the length of the tangent drawn from a point whose distance from the centre of a circle with radius 7 cm is 25 cm. 2. In the given fig., if PQ = PR, prove that QS = RS. P V T T

0 S R 3. If tangents PA and PB from a point P to a circle with centre 0, are incl ined to each other at an angle of 80' then find angle ZPOA. 4. Two circles of radii 10 cm and 8 cm are concentric. Calculate the length of a chord of the larger circle which touches the smaller circle. 5. Two circles touches externally at a point P and from a point T common ta ngent at P, tangent segment TQ and TR are drawn to the two circles. Prove that T Q = TR. 6. A circle is inscribed in AABC having sides 8 cm, 10 cm and 12 cm as shown in fig. Find AD, BE and CF. E F A -12 cm 7. The length of tangents drawn from an external point to the circle are eq ual. Prove it. Use the result to solve the following 98 ?aaedm CCE (Mathematics - X) In the figure, a circle touches all the four sides of a quadrilateral ABCD whose sides AB = 8 cm, BC = 9 cm and CD = 6 cm. Find AD. D B

C DOB A 8. The word tangent came from the Latin word (a) tang (b) tangere (c) axgere (d) tangrant 9. TWo circles Q0, r) and Q0, r2) touching internally such that r, > r2. If thei r centre c, and C2 line on same straight line, then (a) c, - C 2 = r I - r2 (b) c, - C2 = r, (C) C I - C 2 = r2 (d) c, C 2 = r I + r2 10. In figure 0 is centre of a circle, PQ is a chord and the tangent PR at P mak es an angle of 60' with PQ, then ZPOQ is equal to 0 0. 0 (b) 90- (c) 80' (d) 75-

(a) 120' #ANSWERS

TEST YOUR KNOWLEDGE-1 1. 3 cm 3. 12 cm 6. 3 cm 7. 5 cm 8. (b) 2 9. (a) TQ = TR 10. (C) 10 CM TEST YOUR KNOWLEDGE-2 1. 24 cm 3. 500 4. 12 cm 6. AD = 7 cm, BE = 5 cm, CF 3 cm. 7. 5 cm 8. (b) tangere 9. (a) c, - c 2= r, - r, 10.(a) 120* Circles 99 1 1. f FORMAtiVE -ASSESSMENT Activity/Project I Topic Covered : Tangent to the circle. Objective : To find line segment that is tangent-to the circle. )o- Skills Developed Understanding definitions, critical thinking. )o- Time Required: 20 Min. Method Teacher will provide the following sheet to. th~ ~tudents and will ask them to w rite their response. RMUM SO= Name of the student . Identify tangents from the following figures. C A 0 M X X \EB 0 PIN 10 P aS~,N Student's response Circle I AB is tangent to the circle since it touches the circle at one p oint and perpeadiculm to the radius. Circle 11 PT and PQ both are tangents to the circle. Since both touches th e circle at one point and both are perpendicular to the radius.

Circle III NM is not tangent to the circle because NM is not perpendicular to the radius CIM. NS is not tangent to the circle because it intersects the circle in two distinct points. Conclusion I - A tangent touches a circle at exactly one point. pendicular to the radius. Class Work Draw tangent to a circle of radius 5 cm. Homework 2. Tangent is always per

Draw two tangents to a circle from an external point and verify the properties o f tangent to a circle. Quiz Question The point where the tangent and the circle intersects each other is called point of contact. State T/F. Oral Question What is angle between tangent and radius of the circle ? Suggested Activity/Assi gnment Draw two tangents at the end point of the diameter of the circle and verify that these tangents are parallel. 100 -legean a,00 CCE (Mathematics - X) M Activity/Project 2: > Topic Covered : Tangent to the circle. > Objective : To find line segment that is tangent to the circle using t he property that tangent is perpendicular to the radius. > Skills Developed Understanding and application of properties, critical thinking. > Time Required: 20 Min. > Method Teacher will provide the following sheet to the students and wW ask them to writ e their response. Name of the student .

Which of the circles have tangent line, where C is centre of the circle ? B 2 (9!2~(]DS 10 C 15 Student's Response : (11) R (c 3~ M (HI) 50* 30o Q 12 A 5 C 1 13 P

E (C 60' 30* 13~ (IV) Reason Circle Circle Circle Circle I 11 IH W D

M Class Work No PQ No DE tangent is tang6t tangent is tangent

Find the length of tangent to the circle with radius 7 cm ftm pow.. at a distanc e of 17 cin from the centre of circle. e- "':7 Find ZQPO, if PQ is tangent to the circle with centre 0,#M% "It: 70 . 0 T.-r 0 , P M Homework Quiz Qui?stion Defining property of a tangent line is that it intersects the le at 900. State T/F. Oral Question What is length of tangent to a circle fix*n a point at a distance of 3 cm. from the centre of circle with radilis 5 cm. ? Suggested Activity/Assignment -'~ , . . Draw two tangents to a circle from an external point and measure their lengths. . tf Circles 101 F OT I PT [Radius from the point of contact to tangent] In right angled A OTP, Op2 = pT2 + OT2 Op2 = 62 + r2 OPI - r2 = 36 =* OP2 - OA2 = 36 ... Also in right angled AOCA, OA2 = OC2 + AC2 Op2 - (OC2 + AC2) = 36 Op2 - OC2 AC2 = 36

0)

pC2 - AC2 36 (PC - AC) (PC + AC) = 36 AP (PC + BC) = 36 AP (PB) = 36 AP (AP + 5) = 36 [From(i)] [From Op2 - OC2 = pC2] I from centre of circle bisects the chord] => AP2+5AP-36=0 => (AP+9) (AP-4)=O =* AP=4or-9 .'. AP = 4cm. 2. Two tangents PA and PB are drawn to the circle with centre '), such that ZAPB = 120'. Prove that OP = 2AP. Sol. Given. A circle C(O, r). PA and PB are tangents to the circle from P, ou tside the circle such that ZAPB = 120*. OP is joined. A 2 B To Prove. OP = 2PA. Construction. Join OA and OB. Proof. Consider As PAO and PBO PA = PB [Tangents to a circle, from a point outside it, are equal] OP- OP [Common] ZOAP ZOBP AOAP AOBP [RHSI ZOPA = ZOPB 2 In tight angled AOAP, - ZAPB - x 1200 = 600. 2 )0

AP = cos 600 = I =* OP = 2AP. OP 2 E 86 ?&~van *40 ccE (mathematics - x) 3. Prove that the intercept of a tangent between two parallel tangents to a circ le subtends a right angle at the centre. Sol. Given. AB and CD are two tangents to a circle and AB 11 CD. Tangent BD inte rcepts an angle BOD at the centre. A ell 0 Q B

14 P R D To Prove. ZBOD = 90'. Construction. Join OQ and OR. Proof. OP I BD. [A tangent at any point of a circle is perpendicular to the radius through the point of contact] In right angled As OQB and OPB, ZI = Z2, Similarly Z3 = Z4 ZBOD = ZI + Z3 I I - [2ZI + 2Z3)] - (ZI + ZI + Z3 + Z/3) 2 2 1 1 - (ZI + Z2 + Z3 + Z4) = -(180') = 90'. 2 2 4. N all the sides of a parallelogram touch a circle, prove that the parallelogr am is a rhombus. Sol. Given. A circle touches the sides AB, BC, CD and DA of parallelogram ABCD, at P, Q, R and S respectively. D R C YQ A P 8 To Prove. Parallelogram ABCD is a rhombus. Proof. As the tangents to a circle drawn from a point outside the circle are equal in length, .*. AP = AS, BP = BQ; CQ = CR and DR = DS Consider AB + DC = AP + PB + CR + DR = AS + BQ + CQ + DS = (AS + SD) + (BQ + QQ = AD + BC =~AB+AB=AD+AD => 2AB As the . 5. ent to [-.-AB = CD, AD = BC opp. sides of II-] = 2AD =~ AB = AD adjacent sides of the parallelogram ABCD are equal, hence it is a rhombus In the figure, AB is diameter of a circle with centre 0 and QC is a tang the circle at C. If ZCAB = 30', find (i) ZCQA, (ii) ZCBA.

C A (~ ZCQB = 300 ZCQA = 300, ZCBA = 60-. SUMMATIVE ASSESSMENT 1. All the questions of this chapter cover the seven objectives pre scribed by CBSFE Recall 0 Understanding 0 Applying 8 Analysing, Creating 0 Synthesizing 2. These questions also cover three difficulty levels a Easy a Average M High I IMPORTANT MULTIPLE CHOICE QUESTI PRACTICE QUESTIONS 1. If a line intersects a circle in two distinct points, then it is known as a (a) chord (c) tangent (b) secant (d) segment

2. A line segment, having its end point on a circle, is known as (a) chord (c) tangent (b) secant (d) segment

. If a line tou6es the circle at only one point, then it is known as (a) chord (c) tan(yent Z' (b) secant (d) segment

4. Number of tangents that can be drawn through a point on the circle is (a) 3 (C) I (b) 2 (d) 0

I I a Eviduat!" 5. Number of tangents that can be drawn through a point which is inside the circ le is (a) 3 (C) 1

(b) 2 (d) 0 6. Number of tangents that can be drawn through a point which is outside the cir cle is (a) 3 (c) I (b) 2 (d) 0 7. The word tangent came from the latin word (a) tang (c) tangrant (b) iangere (d) axgere 8. The word tangent was introduced by (a) De moivre (c) Disraeli (b) Aryabhata (d) Thomas Fincke Circles 87 9. The common point of the tangent and the circle is called (a) golden point (b) point of contact (c) point of intersection (d) degenrate point 10. A line through point of contact and passing through centre of circle is know n as (a) tangent (c) non-nal (b) chord (d) segment

11. Number of tangents to a circle which are parallel to a secantis (a) 1 (c) 3 (1') 2 (d) infinite

12. A tangent PQ at a point P of a circle of radius 7 cm meets a line through ce ntre 0 at a point Q so that OQ = 25 cm length PQ is (a) 20 cm (c) 24 cm (b) 14 cm (d) 26 cm

13. The length of the segment of the tangent from the external point and the poi nt of contact with the circle is called the length of (a) die chord (c) secant (b) the tangent (d) normal

14. C (0, r,) and C (0, r2) are two concentric circles with ri > r.. AB is a cho rd of C (0, rl) touching C (0, r2) at C then (a) AB = r, (c) AC = BC (b) (d) 15. nce AB = AB = From of Q r 2 r I + r2 a point Q, the length of the tangent to a circle is 12 cm and the dista from the centre is 13 cm. The radius of the circle is (b) 6.5 cm

(a) 7 cm

(c) 5 cm

(d) 9 cm

16. TP and TQ are the two tangents to a circle with centre 0, so that ZPOQ = 100 ', then ZPTQ is equal to (a) 60' (b) 70 (C) 80' (d) 9017. TP and TQ are two tangents to a circle with centre 0 so that ZPOQ = 120', th en ZOPT is equal to (a) 50' (C) 80' (b) 60 (d) 9018. Two concentric circles are of radii 13 cm and 5 cm. The length of the chord of larger circle which touches the smaller circle is (a) 12 cm (c) 24 cm (b) 20 cm (d) 26 cm 19. A quadrilateral ABCD i. drawn to circumscribe a circle. If AB = 1i cm, BC = 15 cm and CD = 14 cm, then AD is equal to (a) 10 cm (c) 12 cm (b) (d) 20. and I I cm 14 cm A triangle ABC is drawn to circumscribe a circle. If AB = 13 cm, BC = 14 cm AE = 7 cm, then AC is equal to

88 7ageffem a4WO CCE (Mathemati cs - X~ (a) 12 cm (C) I I cm A AE B D C (b) 15 cm (d) 16 cm 21. A right AABC right angled at A drawn to circumscribe a circle of radius 5 cm with centre 0. If AC = 17 cm and AB = 18 cm, then OC is equal to (a) 10 cm (c) 12 cm (b) 9 cm (d) 13 cm 22. A circle is inscribed in a triangle with sides 8, 15 and 17 cm. The radius o f the circle is (a) 6 cm (c) 4 cm (b) 5 cm (d) 3 cm 23. Distance between two parallel lines is 14 cm. The radius of circle which wil l touch both two lines is (a) 6 cm (c) 12 cm (b) 7 cm (d) 14 cm 24. A line Im' is tangent to a circle with radius 5 cm. Distance between the centre of circle and the line M is (a) 3 cm (c) 5 cm (b) 4 cm (d) 6 cm 25. A line touches a circle of radius 4 cm. Another line is drawn which is tange

nt to the circle. If the two lines are parallel then distance between them is (a) 4 cm (c) 7 cm (b) 6 cm (d) 8 cm 26. Two parallel lines touch the circle at points A and B respectively. If area of the circle is 25 X CM2, then AB is equal to (a) 5 cm (C) 10 cm 27. Match the column: (b) 8 cm (d) 25 cm (1) The tangent at any (A) Known as a tangent point of a circle is to the circle (2) The line containing the (B) Perpendicular to the radius through the radius through the point of contact is point of contact (3) The lengths of tangents (C) Called the 'normal' drawn from an external to the circle point to a circle are (4) When two end points of (D) Equal the corresponding chord of a secant coincide, it is (a) I ) A, 2 B, 3 C, 4 D (b) I --- > B9 2 A, 3 D, 4 C (c) 1 -4 D, 2 A, 3 --- ) C, 4 B (d) I ---) B, 2 C, 3 -4 D, 4 A MuL-nPLE CHOICE OUESTIONS 1. In Fig. if ZAOB = 125', then ZCOD is equal to (a) 62.5' (b) 45 (c) 35' (d) 55A 125' 1`1& B D C 2. In Fig. AB is a chord of the circle and AOC is its diameter such that -/ ACB = 50'. If AT is the tangent to the circle at the point A, then ZBAT is equal to (a) 6 5' (b) 60 (c) 50' (d) 40CO I 50) B - * A T 3. From a point P which is at a distance of 13 cm from the centre. 0 of a c ircle of radius 5 cm, the pair of tangents PQ and FIR to the circle are drawn. T hen the area of the quadrilateral PQOR is (a) 60 cm' (c) 30 cm' (b) 65 cm2 (d)32.5 CM2

28. A circle touches x-axis at A and y-axis at B. If 0 is origin and OA = 5 units, then diameter of the circle is (a) 8 units (c) 10-,f2- units NCERT EXEMPLAR PROBLEMS I (b) 10 units (d) 8-vF2 units 4. In Fig. AT is a tangent to the circle with centre 0 such that OT = 4 cm. and ZOTA = 30'. Then AT is equal to (a) 4 cm (c) 2-,f3- cm 4 (b) 2 cm (d) 4 -. 5- C m CM 30 5. In Fig. if 0 is centre of a circle, PQ is a chord and the tangent PR at P mak es an angle of 50' with PQ, then ZPOQ is equal to (a) 100' (C) 90' (b) 80(d) 75P 50 0 70 6. R In Fig. if PQR is the tangent to a circle at Q whose centre is 0, AB is a chord parallel to PR and.ZBQR = 70', then ZAQB is equal to (a) 20' (c) 35' B

0

A 0, 'P (2o

SUB)ECTIVE QUESTIO (b) 40(d) 45I VERY SHORT ANSWER TYPE QUESTIONS ] PREVIOUS YEARS' QUESTIONS (a) In fig., if ZATO = 40', find ZAOB. AA 0 B>T z- [Al 20081

(b) From a point P, the length of the tangent to a circle is 15 cm and dista nce of P from the centre of the circle is 17 cm. Then what is the radius of the circle ? [Delhi 2008 Q (c) The two tangents from an external point P to a circle with centre 0 are PA and PB. If ZAPB = 70', what is the value of ZAOB ? JAI 2008 C1 Circles 89 ~ (d) In figure, CP and CQ are tangents to a circle with centre 0. ARB is anot her tangent touching the circle at R. If CP = I I cm, and BC = 7 cm, then find t he length of BR. p A 00 Q Q R I B C

[Delhi 20091 (e) In figure, AABC is circumscribing a circle. Find the length of BC. T 4 cm ~ R 3 cm A Q 11 CM . b B \A C [AT 20091 (f) In figure, CP and CQ are tangents from an external point C to a circle w ith centre 0. AB is another tangent which touches the circle at R. If CP = 11 cm and BR 4 cm, find the length of BC. P C)A R 00 Q [AT 20101 (g) A tangent PQ at a point P of a circle of radius 5 cm meets a line throug h the centre 0 at a point Q so that OQ = 13 cm. Find the length PQ. [Foreign 201 0] PRACTICE QUESTIONS \

P

C B

2. (a) A tangent PQ at a point P to a circle of radius 6 cm meets a line throug h the centre 0 at a point Q so that OQ = 10 cm., find the length of PQ (b) From a point Q, the length of the tangent to a circle is 24 cm and the d istance of Q from the centre is 25 cm., find the radius of the circle. 3. In the giv en fig., if TP and TQ are the two tangents to a circle with centre 0 so that ZPOQ = 140', find ZFTQ. P 140. #Q 90 %gean aWA- CCE (Mathematics - X) 5. If tangents PA and PB from a point P to a circle with centre 0 are inclined to e ach other at angle of 60', find ZPOA. A point P is 13 cm from the centre of the circle. The length of the tangent draw n from P to the circle is 12 cm. Find the radius of the circle. 6. Find the length of the tangent drawn from a point whose distance from th e centre of a circle is 25 cm. Given that radius of the circle is 7 cm. 7. If OL = 5 cm., OA = 13 cm., find AB ? 0 0 E '!~e 0 LO AWL 8. TAP 10 C Q D 9. In the given fig., OD is perpendicular to the chord AB of a circle whose centre is 0. If TIC is a diameter, find CA = ? OD B A Z D 0 C ~r 10. Yes/No 11. 12. Given an arc of a circle, is there a method to complete the circle ? Yes/No A quadrilateral ABCD is drawn to circumscribe a circle as shown in the fig., then AB + CD = ?

B In the fig., 0 is the centre of the circle with radius 5 cm, AB I I CD, AB = 6 cm. Find OP. B

Is there a method to find the centre of a given circle ?

S A D ( R P a 13. Two concentric circles are of radii 5 cm and 3 cm. Find out the length o f the chord of larger circle which touches the smaller circle. 14. In the fig., ABCD is a cyclic quadrilateral and PQ is tangent to the cir cle at C. If BD is a diameter, Z-DCQ = 40' and ZABD = 60', find ZBCP. i 0 D 4\0 - C 0 6 \0' 15. 16. How many parallel tangents can a circle have ? Wh,-'; :the distance between two parallel tangents of a circle of radius 7 cm. C B

17. PQ isa tao9zent drawn from a point Pto a circle with centre 0. and Q01, s a diameter of the circle such that ZPOR I 10'. Find ZOPQ. I SHORT ANSWER TYPE QUESTIONS-1] PREVIOUS YEARS' OUESTIONS 1. B 0 OR 80 4 P A a In figure, 0 is the centre of the circle, PQ is a tangent to the circle at A. If ZPAB = 58', find ZABQ and ZAQB.

JAI 1999C] 2. In figure, a circle touches the side BC of AABC at P and touches AB and AC pr oduced at Q and R respectively. If AQ = 5 cm, find the perimeter of AABC. A ABE C Q XR [Delhi 20001 3. A tangent PT is drawn parallel to a chord AB as shown in figure. Prove that APB is an isosceles triangle. [Foreign 2000] T P

r/71 .0 +0 18. From an external point P, k tangents can be drawn to a circle. Find the value of k. 19. In the figure given below, find ZQSR. a S 0 T 50- P R E:~ 20. In the given figure AB, AC and AD are tangents of AB = 5 cm, find AD. A B 10 C, D 00 C 0 P b 4. (a) In figure, XP and XQ are two tangents to a circle with centre 0 from a po int X outside the circle. ARB is tangent to circle at R. Prove that XA + AR = XB + BR. P A X 0 0 B Q [Delhi 20031 (b) Two tangents PA and PB are drawn to a circle with centre 0 from an external point P. Prove that ZAPB = 2 ZOAB. A

P 0 [Delhi 20031 B (c) Prove that the parallelogram circumscribing a is a rhombus. [Delhi 2009] (d) If all the sides of a parallelogram touch that the parallelogram is a rhombus. [Delhi 20101 (e) In figure, there are two concentric circles, with 5 cm and 3 cm. From an external point P, tangents PA and circles. If AP = 12 cm, find the length of BP. A 0. B JAI 20101 Circles 91 1 NCERT QUESTIONS 5. In the given figure, TAS is a tangent to the circle, with centre 0, at the point A. If ZOBA = 32', find the value of x.

circle a circle, show ceiitre 0 and of radii PB are drawn to these

T A S a0 x 2' 6. In the given figure, ABC is a right-angled triangle, fight angled at A, with AB = 6 cm and AC = 8 cm. A circle with centre 0 has been inscribed insid e the triangle. Calculate the value of r, the radius of the inscribed circle. C C E 0 co r r A 6 cm 7. In the given figure, PT is tangent to the circle at T. If PA 4 cm and AB = 5 cm, find PT. ZB A P T P is the mid-point of an arc QPR of a circle. Show that the tangent at P is para llel to the chord QR. 9. If AABC is isosceles with AB = AC, and C(O, r) is the incircle of AABC t ouching BC at L. Prove that the point L bisects BC. 10. If AABC is isosceles with AB = AC, prove that the tangent at A to the circumcircle of AABC is parallel to BC. 11. Two circles touch intemally at a point P and from a point T the common t angent at P, tangent segments TQ, TR are drawn to the two circles. Prove that TQ = TR. 12. In the given figure, 0 is the centre of the circle. Determine ZAOB and ZAMB, if PA and PB are tangents. 8~ PREVIOUS YEARS' QUESTIONIII 1. In figure, AB and CD are two parallel tangents to a circle with centre 0 . ST is tangent segment between the two parallel tangents touching the circle at Q. Show that ZSOT = 900. . A 92 ImpaAm aWA- CCE (Mathematics - X) P