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Class 10 Mathematics by VP Classes

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NCERT TEXTBOOK QUESTIONS SOLVED
Page No. 213
EXERCISE 10.2
In Q.1 to 3, choose the correct option and give justification.

Q.1. 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
Sol. From figure
 Notes | EduRev 
OQ2 = OP2 + PQ2
(25)2 = OP+ (24)2
⇒ 625 - 576 = OP2
⇒ 49 = OP2
 Notes | EduRev
OP = 7cm
Radius of the circle = 7cm.
Hence, correct option is (a).

Q.2. In 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°

 Notes | EduRev
Sol. ∠OPT = 90°; ∠OQT = 90°; ∠POQ = 1100
TPOQ is a quadrilateral
⇒ ∠PTQ + ∠POQ = 180°
⇒ ∠PTQ + 1100 = 1800
⇒ ∠PTQ  = 1800 - 1000 = 700

Q.3. Choose the correct option: 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°
Sol. In ΔOAP and ΔOBP
OA = OB    [Radii]
PA = PB
[Length of tangents from an external point are equal]
OP = OP     [common]
 Notes | EduRev
 Notes | EduRev
 Notes | EduRev
Hence, Option (a) is correct

Page No. 214
Q.4. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Sol. AB is a diameter of the circle, p and q are two tangents
OA ⊥ p and OB ⊥ q and ∠1 = ∠2 = 900
⇒ p ║q    [∠1 and ∠2 are alternate angle]
 Notes | EduRev

Q.5. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Sol. XY tangent to the circle C(O, r) at B and AB ⊥ XY
 Notes | EduRev
Join OB
∠ABY = 90º   
∠OBY = 90º
[Radius through point of contact is ⊥ to the tangent]
∴ ∠ABY + ∠OBY = 1800
⇒ ABO is collinear.
∴ AB passes through centre of the circle.

Q.6. 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.
Sol. OP = Radius of the circle
OA = 5 cm; AP = 4 cm
 Notes | EduRev

∠OPA = 90°   [Radius and tangent are ⊥ar]
OA2 = AP2 + OP2    [By Pythagoras theorem]
52 = 42 + OP2 
⇒ 25 = 16 + OP2
⇒ 25 - 16 = OP2
⇒ 9 = OP2
⇒ OP = √ 9 = 3
Radius = 3 cm

Q.7. 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.
Sol. The radius of larger circle = 5 cm and radius of smaller circle = 3 cm
OP ⊥ AB    [Radius of circle is perpendicular to the tangent]
AB is a chord of the larger circle
∴ OP bisect  AB   AP = BP
In ΔOAP     OA2 =    AP2 + OP2
    (5)2 =   AP2 + (3)2
    AP2 =   25 - 9 = 16
   Notes | EduRev
AB = 2AP =   2 x 4 = 8 cm

Q.8. A quadrilateral ABCD is drawn to circumscribe a circle (see figure). Prove that: AB + CD = AD + BC

 Notes | EduRev

Sol. 
AP = AS    ...(i)    [Lengths of tangents from an external point are equal]
BP = BQ    ...(ii)
CR = CQ ...(iii)
DR = DS   ...(iv)
Adding equations (i), (ii), (iii) and (iv), we get
AP + BP + CR + DR = AS + BQ + CQ + DS
⇒ (AP + BP) + (CR + DR) = (AS + DS) +(BQ + CQ
 AB + CD = AD + BC

Q.9. In the 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°.

 Notes | EduRev
Sol. Given: Two parallel tangents to a circle with centre O. Tangent AB with point of contact C intersects XY at A and X'Y' at B
To Prove: ∠AOB = 90°
Construction: Join OA, OB and OC
Proof: In ΔAOP and ΔAOC
AP = AC [Lengths of tangents]
OP = OC [Radii]
OA = OA
⇒ ΔAOP ≅ ΔAOC   [SSS congruence rule]
⇒ ∠PAO = ∠CAO    [C.P.C.T]
∠PAC = 2 ∠OAC    ...(i)
Similarly ∠QBC = 2 ∠OBC    ...(ii)
Adding (i) and (ii), we get
∠PAC + ∠QBC = 2      [∠OAC + ∠OBC]
∠PAC + ∠QBC = 180°
[interior consecutive angle on same side of transversal]
2 = 180° [∠OAC + ∠OBC]
⇒ ∠OAC + ∠OBC = 90°
In ΔAOB, ∠AOB + [∠OAC + ∠OBC] = 180°
⇒ ∠AOB + 90° = 180°
⇒ ∠AOB = 90°

Q 10. 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.
Sol. PA and PB are two tangents, A and B are the points of contact of the tangents.
OA ⊥ AP and OB ⊥ BP
∠OAP = ∠OBP = 90°
[Radius and tangent are perpendicular to each other]
In quadrilateral OAPB
(∠OAP + ∠OBP) + ∠APB + ∠AOB = 3600
 Notes | EduRev
⇒ 180° + ∠APB + ∠AOB = 360°
∠APB + ∠AOB = 360° - 180° = 180°

Q 11. Prove that the parallelogram circumscribing a circle is a rhombus. 
Sol. Parallelogram ABCD circumscribing a circle with centre O.
 Notes | EduRev
OP⊥ AB and OS ⊥ AD
In ΔOPB and ΔOSD, ∠OPB = ∠OSD [Each 90°]
OB = OD
[Diagonals of ║gm bisect each other]
OP = OS    [Radii]
  ΔOPB ≅ ΔOSD [RHS congruence rule]
PB = SD     [C.P.C.T] ...(i)
AP = AS [Lengths of tangents]... (ii)
Adding (i) and (ii)
AP + PB = AS + DS
   AB AD
Similarly  AB = BC = CD = DA
∴ ║gm ABCD is a rhombus

Q 12. 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 figure). Find the sides AB and AC.
 Notes | EduRev
Sol. BD = 8 cm and DC = 6 cm
BE = BD = 8 cm; CD= CF = 6 cm;
 Notes | EduRev
Let AE=AF = x cm
In ΔABC   a = 6 + 8 = 14 cm;
b = (x + 6) cm;
c = (x +8)
 Notes | EduRev
 Notes | EduRev
ar ΔABC = ar ΔOBC + ar ΔOCA + ar ΔOAB
 Notes | EduRev
From (i) and (ii)
 Notes | EduRev
⇒ 3x(x+14) = (x + 14)2 
⇒ 3x2 + 42x = x2 + 196 + 28x
⇒ 2x2 + 14x - 196 = 0
⇒ x2 + 7x - 98 = 0
⇒ x2 + 14x - 7x - 98 = 0
⇒ x(x + 14) - 7(x + 14) = 0
⇒ (x - 7)(x + 14) = 0 ⇒ x = 7
AB = x + 8 = 7 + 8 = 15 cm
AC = x + 6 = 7 + 6 = 13 cm

Q 13. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Sol. 
 Notes | EduRev

AB touches at P and BC, CD and DA touch the circle at Q, R and S.
Construction: Join OA, OB, OC, OD and OP, OO, OR, OS
∴ ∠1 = ∠2    [OA bisects ∠POS]
Similarly
 Notes | EduRev
 Notes | EduRev
Similarly ∠AOB + ∠COD = 180°
Hence, opposite sides of quadrilateral circumscribing a circle subtend supplementary angles at the centre of a circle.

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