Ex. 1Multiple choice
If angle between two radii of a circle is 130º, the angle between the tangents at the ends of the radii is :
- (A)90º
- (B)50º
- (C)70º
- (D)40º
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Solution : Answer (B)
Chapter 9 – Circles
Class 10 Mathematics · 46 questions · 23 with answers
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Ex. 2Multiple choice
In Fig. 9.1, the pair of tangents AP and AQ drawn from an external point A to a circle with centre O are perpendicular to each other and length of each tangent is 5 cm. Then the radius of the circle is
- (A)10 cm
- (B)7.5 cm
- (C)5 cm
- (D)2.5 cm
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Solution: Answer (C)
Ex. 3Multiple choice
In Fig. 9.2, PQ is a chord of a circle and PT is the tangent at P such that ∠QPT = 60°. Then ∠PRQ is equal
- (A)135°
- (B)150°
- (C)120°
- (D)110°
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Solution : Answer (C) [Hint : ∠OPQ = ∠OQP = 30°, i.e., ∠POQ = 120°. Also, ∠PRQ = reflex ∠POQ]
Ex. 1Short answerExercise 9.1
In Fig. 9.9, BOA is a diameter of a circle and the tangent at a point P meets BA extended at T. If ∠PBO = 30°, then ∠PTA is equal to 30°.
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Solution : True. As ∠BPA = 90°, ∠PAB = ∠OPA = 60°. Also, OP⊥PT. Therefore, ∠APT = 30° and ∠PTA = 60° – 30° = 30°. CIRCLES 105
Ex. 2Short answerExercise 9.1
In Fig. 9.10, PQL and PRM are tangents to the circle with centre O at the points Q and R, respectively and S is a point on the circle such that ∠SQL = 50° and ∠SRM = 60°. Then ∠QSR is equal to 40°.
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Solution : False. Here ∠OSQ = ∠OQS = 90°–50° = 40° and ∠RSO = ∠SRO = 90° – 60° = 30°. Therefore, ∠QSR = 40° + 30° = 70°.
Ex. 1Short answerExercise 9.2
If d1, d2 (d2 > d1) be the diameters of two concentric circles and c be the length of a chord of a circle which is tangent to the other circle, prove that d22 = c2 + d12.
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Solution : Let AB be a chord of a circle which touches the other circle at C. Then ∆OCB is right triangle (see Fig.9.11). By Pythagoras theorem OC2 +CB2 = OB2. 2 2 2
Ex. 2Short answerExercise 9.2
If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by a + b − c r = .
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Solution : Let the circle touches the sides BC, CA, AB of the right triangle ABC at D, E and F respectively, where BC = a, CA = b and AB = c (see Fig. 9.12). Then AE = AF and BD = BF. Also CE = CD = r. i.e., b – r = AF, a – r = BF or AB = c = AF + BF = b – r + a – r CIRCLES 107 a + b − c This gives r =
Ex. 1Multiple choiceExercise 9.3
In Fig. 9.15, from an external point P, a tangent PT and a line segment PAB is drawn to a circle with centre O. ON is perpendicular on the chord AB. Prove that :
- (i)PA . PB = PN2 – AN2
- (ii)PN2 – AN2 = OP2 – OT2
- (iii)PA.PB = PT2
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Solution : (i) PA . PB = (PN – AN) (PN + BN) = (PN – AN) (PN + AN) (As AN = BN) 2 2 = PN – AN (ii) PN2 – AN2 = (OP2 – ON2) – AN2 (As ON⊥PN) = OP2 – (ON2 + AN2) = OP2 – OA2 (As ON⊥AN) = OP2 – OT2 (As OA = OT) CIRCLES 109 (iii) From (i) and (ii) PA.PB = OP2 – OT2 = PT2 (As ∠OTP = 90°)
Ex. 2Short answerExercise 9.3
If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that AQ = (BC + CA + AB)
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Solution : See Fig. 9.16. By Theorem 10.2 of the textbook, BQ = BP CP = CR, and AQ = AR Now, 2AQ = AQ + AR = (AB + BQ) + (AC + CR) = AB + BP + AC + CP = (BP + CP) + AC + AB = BC + CA + AB i.e., AQ = (BC + CA + AB).
Questions
If radii of two concentric circles are 4 cm and 5 cm, then the length of each chord of one circle which is tangent to the other circle is
- (A)3 cm
- (B)6 cm
- (C)9 cm
- (D)1 cm
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(B) 6 cm
In Fig. 9.3, if ∠AOB = 125°, then ∠COD is equal to
This question refers to a figure in the original PDF.
- (A)62.5°
- (B)45°
- (C)35°
- (D)55°
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(D) 55°
In Fig. 9.4, 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 ∠BAT is equal to
This question refers to a figure in the original PDF.
- (A)65°
- (B)60°
- (C)50°
- (D)40° CIRCLES 103
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(C) 50°
From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is
- (A)60 cm2
- (B)65 cm2
- (C)30 cm2
- (D)32.5 cm2
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(A) 60 cm2
At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A is
- (A)4 cm
- (B)5 cm
- (C)6 cm
- (D)8 cm
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(D) 8 cm
In Fig. 9.5, AT is a tangent to the circle with centre O such that OT = 4 cm and ∠OTA = 30°. Then AT is equal to
This question refers to a figure in the original PDF.
- (A)4 cm
- (B)2 cm
- (C)2 3 cm
- (D)4 3 cm
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(C) 2 3 cm
In Fig. 9.6, if O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to
This question refers to a figure in the original PDF.
- (A)100°
- (B)80°
- (C)90°
- (D)75°
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(A) 100°
In Fig. 9.7, if PA and PB are tangents to the circle with centre O such that ∠APB = 50°, then ∠OAB is equal
This question refers to a figure in the original PDF.
- (A)25°
- (B)30°
- (C)40°
- (D)50°
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(A) 25°
If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to
- (A)3 cm
- (B)6 cm
- (C)3 cm
- (D)3 3 cm
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(D) 3 3 cm
In Fig. 9.8, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to
This question refers to a figure in the original PDF.
- (A)20°
- (B)40°
- (C)35°
- (D)45°
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(B) 40°
If a chord AB subtends an angle of 60° at the centre of a circle, then angle between the tangents at A and B is also 60°.
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False
The length of tangent from an external point on a circle is always greater than the radius of the circle.
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False
The length of tangent from an external point P on a circle with centre O is always less than OP.
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True
The angle between two tangents to a circle may be 0°.
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True
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = a 2 .
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True
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = a 3 .
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False
The tangent to the circumcircle of an isosceles triangle ABC at A, in which AB = AC, is parallel to BC.
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True
If a number of circles touch a given line segment PQ at a point A, then their centres lie on the perpendicular bisector of PQ.
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False
If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of PQ.
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True
AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD.
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True
1 1 i.e., d1 + c = d2
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False
2 2 (As C bisects AB) Therefore, d22 = c2 + d12.
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False
Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.
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3 cm
Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral.
If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that ∠DBC = 120°, prove that BC + BD = BO, i.e., BO = 2BC.
Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
In Fig. 9.13, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD.
This question refers to a figure in the original PDF.
In Question 5 above, if radii of the two circles are equal, prove that AB = CD.
In Fig. 9.14, common tangents AB and CD to two circles intersect at E. Prove that AB = CD.
This question refers to a figure in the original PDF.
A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.
Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.
Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A.
If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC + DE + FA.
Let s denote the semi-perimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively, prove that BD = s – b.
From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the the perimeter of the triangle PCD.
If AB is a chord of a circle with centre O, AOC is a diameter and AT is the tangent at A as shown in Fig. 9.17. Prove that ∠BAT = ∠ACB .
This question refers to a figure in the original PDF.
Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.
In a right triangle ABC in which ∠B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P bisects BC.
In Fig. 9.18, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS. CIRCLES 111 [Hint: Draw a line through Q and perpendicular to QP.]
This question refers to a figure in the original PDF.
AB is a diameter and AC is a chord of a circle with centre O such that ∠BAC = 30°. The tangent at C intersects extended AB at a point D. Prove that BC = BD.
Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.
In Fig. 9.19, the common tangent, AB and CD to two circles with centres O and O' intersect at E. Prove that the points O, E, O' are collinear.
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In Fig. 9.20. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB.
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The tangent at a point C of a circle and a diameter AB when extended intersect at P. If ∠PCA =110º , find ∠ CBA [see Fig. 9.21]. [Hint: Join C with centre O.]
This question refers to a figure in the original PDF.
If an isosceles triangle ABC, in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, find the area of the triangle.
A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ∆ABC.