Class 10 Maths Extra Questions Chapter 10 Circles

14. Choose the correct Answer.
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
15. A circle can have _______ parallel tangents at the most.
16. The common point of a tangent to a circle and radius of the circle is called
________.

SHORT ANSWER TYPE QUESTIONS

17. If diameters of two concentric circles are d1 and d2 (d2 > d1) and c is the
length of chord of bigger circle which is tangent to the smaller circle. Show
that d22 = c2 + d12.

18. The length of tangent to a circle of radius 2.5 cm from an external point P is 6
cm. Find the distance of P from the nearest point of the circle.

19. TP and TQ are the tangents from the external point T of a circle with centre O.
If OPQ = 30° then find the measure of TQP.

20. In the given fig. AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semi perimeter
of ABC.

A
4c
m

R P

C Q B
6 cm
21. A circle is drawn inside a right angled triangle whose sides are a, b, c where c
is the hypotenuse, which touches all the sides of the triangle. Prove
a+b–c
r= where r is the radius of the circle.
2
Mathematics-X 143

22. Prove that the tangent at any point of a circle is perpendicular to the radius
through the point of contact.
23. Prove that in two concentric circles the chord of the larger circle which is
tangent to the smaller circle is bisected at the point of contact.
24. In the given Fig., AC is diameter of the circle with centre O and A is the point
of contact, then find x.

C

x

O B

40°
P Q
A
25. In the given fig. KN, PA and PB are tangents to the circle. Prove that:
KN = AK + BN.

26. In the given fig. PQ is a chord of length 6 cm and the radius of the circle is 6
cm. TP and TQ are two tangents drawn from an external point T. Find PTQ.

P

O T

Q

144 Mathematics-X

LONG ANSWER TYPE QUESTIONS
27. In the given figure find AD, BE, CF where AB = 12 cm, BC = 8 cm and
AC = 10 cm.

C

F E

A B
D

28. Two tangents PA and PB are drawn to a circle with centre O from an external
point P. Prove that APB = 2 OAB (NCERT, Exemplar)

A

O

P
B
29. In the given fig. OP is equal to the diameter of the circle with centre O. Prove
that ABP is an equilateral triangle.

A

o P

B

Mathematics-X 145

30. In the given fig., find PC. If AB = 13 cm, BC = 7 cm and AD = 15 cm.
A

R
B
O 4 cm
S Q

C P D
31. In the given figure, find the radius of the circle.

A
cm
23

R

B
5 cm 29 cm
O r
S Q

C P D

32. In the given fig. PQ is tangent and PB is diameter. Find the values of angles x
and y.
P
y
x y
A o Q
35°

B

146 Mathematics-X

33. In given figure, two circles touch each other at the point C. Prove that the
common tangent to the circles at C, bisects the common tangent at P and Q.

P

T
A
Q
C
B

34. In the given figure, a circle touches all the four sides of a quadrilateral ABCD.
If AB = 6 cm, BC = 9 cm and CD = 8 cm, then find the length of AD.

C
D

B
A

35. In figure, PA is a tangent from an external point P to a circle with centre O, If
POB = 115°. Find APO.

A

P O
115°

B

36. 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.

Mathematics-X 147

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

38. In the given figure, find the perimeter of ABC, if AP = 12 cm.

A

D
B C
P Q

39. In the given figure, a quardrilateral ABCD is drawn to circumscribe a circle.
Prove that AB + CD = BC + AD
A B

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

148 Mathematics-X

ANSWERS AND HINTS

1. Since length of both the tangents from a point outside the circle is equal, So
BN = BL, CM = CL
BL + CL = BC = 10 cm

2.

By Pythagoras Theorem, QR = 7 cm.
3. Angle in the same segment are euqal.
 DC is the chord so DAC = DBC = 60°.
 The sum of the opposite angles of a cyclic quadrilateral is 180°.
So BCD = 70°
4. The tangent at any point of a circle is perpendicular to the radius through the
point of contact.
So, RPO = 90°
OPQ = OQP = 40°
POQ = 100°
Q

3 cm
60°
O P
5.
3 cm

R
QPO  RPO
60
 QPO = RPO   30
2
In QPO, OQP = 90°(Tangent is perpendicular at the point of contact).
OQ
tan 30° =  QP = 3 3 cm
QP

Mathematics-X 149

6.

In AOP, right angled at P.
OA2 = AP2 + OP2  (5)2 = AP2 + 42  AP2 = 9
 AP = 3
 AB = 6 cm (  OP  AB so OP bisects AB)
7. In PQO (4)2 + (3)2 = (OP)2
5 = OP
In PRO, (5)2 = (2)2 + (PR)2
PR = 21 cm
A

4
Q 5 3 40° 1 P
8. O
2

B
In Quadrilateral OAPB
1 + 2 + 3 + 4 = 360°
1 + 3 = 180°
3 = 140°
Now, 3 = 2 5
5 = 70° or AQB = 70°

1  2 
3  4 
9.
5  6  (CPCT) of their corresponding triangles.

7  8
°
2 ( 2  3  6  7) =360°
or AOB + COD = 180° 6
or COD = 55°
150 Mathematics-X

10. OQT = 90° (Angle between tangent & radius)
PQO = 30°
PQO = OPQ = 30°
11. Infinitely many
12. One
13. 90°

14. d( 119 cm)
15. Two
16. Point of Contact

17.

AB = c
AO2 = OP2 + AP2
2 2
 d2  d 
  =  1   AP 2
2  2
2 2
 d2   d1 
     = AP2
2 2
1
(d 2 )  (d1) 
2 2
= AP
4
1
(d 2 )  (d1 )  = AB
2 2
2
4
(d 2 )2  (d1 )2 = c
2
(d 2 )2  (d1 )2 = c
d 2 2 = c2 + d 2
1

18.

Mathematics-X 151

(OP)2 = (OT)2 + (PT)2
(OP)2 = (2.5)2 + (6)2
= 42.25
(OP)2 = (6.5)2  OP = 6.5 cm
QP = 4 cm

19.

OQP = OPQ = 30°
OQT = 90° (Angle between radius and tangent)
TQP = OQT – OQP
= 90° – 30° = 60°
20. AP = AR = 4 cm
CR = CQ = (9 – 4) cm = 5 cm
1
Semi perimeter = [AC + AB + BC]
2
1
=
2
 9  10  11 = 15 cm
A

21.
c
F b

o r
E

B D C
a

152 Mathematics-X

b – r = AF, a – r = BF
or, AB = c = AF + BF = b – r + a – r
a+b–c
This gives, r=
2
23. Join OP
AB is tangent to circle C1 at P and OP is radius C1 C2
OP  AB O
AB is chord of circle C2 and OP  AB.
A B
P
Therefore OP is the bisector of the chord AB as the
perpendicular from the centre bisects the chord i.e,
AP = BP
24. OAB = 50°
x + B + OAB = 180°
x + 90° + 50° = 180°
x = 40°
25. AK = KC
BN = NC
 KN = KC + NC =AK + BN
26. POQ + PTQ = 180°
60° + PTQ = 180°
PTQ = 120°
27. AC = AF + FC = 10 cm …(1)
AB = AD + DB = 12 cm …(2)
BC = BE + CE = 8 cm …(3)
 BD = BE 
 AD = AF 
 CF = CE  …(4)
 
AC = AD + FC = 10 cm …(5)
AB = AD + DB = 12 cm …(6)
BC = BD + CF = 8 cm …(7)

Mathematics-X 153

Add (5, 6, 7)
2(AD + FC + DB) = 30
AD + FC + DB = 15
Substitute values from (1), (2) & (3)
and find. AD = 7 cm, BE = 5 cm, CF = 3 cm. A
28. PA = PB
1 2 O
So, 2 = 3 = (180  1) 4
3
2 1
1 P
2 = 3 = 90  1 B
2
4 = 90° (Angle between tangent & Radius)
OAB = 4 – 2
 1 
= 90   90  1
2
1
 OAB = APB
2
OAB = APB
29. OP = 2r
 OQ = QP = r
A

O P
Q

B
Consider AOP in which OA  AP and OP is the hypotenuse.
OQ = AQ = OA
(Mid point of hypotenuse is equidistance from the vertices).
 OAQ is an equilitateral triangle.
 AOQ = 60°
Consider right angled triangle OAP
AOQ = 60°
OAP = 90°  APO = 30°
APB = APO = 2 × 30° = 60°
PA = PB (tangents)

154 Mathematics-X

 PAB = PBA
APB = 60°
180  60
PAB = PBA =  60
2
 ABP is an equilateral triangle.
30. PC = 5 cm
31. 11 cm
32.

In ABP, 1 = 90° (Angle in semi-circle)
1 + 35° + y = 180°
90° + 35° + y = 180°
y = 55°
OPQ, 2 = 90° (Angle between tangent and radius)
2 + x + y = 180°
90° + x +  = 180°
x = 35°
34. AD = 5 cm
35. 25°
38. 24 cm

Mathematics-X 155

PRACTICE-TEST
CIRCLES
Time : 1 Hr. M.M.: 20

SECTION-A
1. In the given figure find x, where ST is the tangent. 1
T

x-40° x
S
O

2. In the given figure if AC = 9 cm, find BD. 1

A B C

D

3. In the given figure, ABC is circumscribing a circle, then find the length of
BC.
1

A
m
3c

N M
8 cm
m
4c

B C
L
4. From the external point P, tangents PA and PB are drawn to a circle with centre
O. If PAB = 50°, then find AOB. 1

156 Mathematics-X

SECTION-B
5. If the angle between two tangents drawn from an external point P to a circle of
radius a and centre O is 60° then find the length of OP. (All India 2017) 2
6. In the following figure find x. 2

7. Two concentric circle with centre O are of radii 6 cm and 3 cm. From an external
point P, tangents PA and PB are drawn to these circle as shown in the figure. If
AP = 10 cm. Find BP 2

A

o P

SECTION-C
8. In the given figure, AB is a tangent to a circle with centre O. Prove
BPQ = PRQ. 3
R

O Q

A P B
Mathematics-X 157

9. In the given figure ABC is drawn to circumscribe a circle of radius 3 cm,
such that the segment BD and DC into which BC is divided by the point of
contact D are of length 6 cm and 8 cm respectively, find side AB if the ar(ABC)
= 63 cm² 3

A

F E

B C
6 cm D 8 cm

SECTION-D
10. AB is a diameter of a circle with centre O and AT is a tangent. If AOQ = 58°
find ATQ. 4

158 Mathematics-X