Page 1 :
Circles, 2016, Very Short Answer Type Questions [1 Mark], , Question 1., From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB =, 50°, then find ∠AOB., Solution:, , Question 2., In given figure, PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and, ∠CAB = 30°, find ∠PCA, , Solution:
Page 2 :
Question 3., In figure given, AOB is a diameter of a circle with centre O and AC is a tangent to the circle, at A. If ∠BOC = 130°, then find ∠ACO., , Solution:, , Short Answer Type Questions I [2 Marks], Question 4., In given figure, a circle is inscribed in a ΔABC, such that it touches the sides AB, BC and CA, at points D, E and F respectively. If the lengths of sides AB, BC and CA are 12 cm, 8 cm and, 10 cm respectively, find the lengths of AD, BE and CF, , Solution:, , Question 5., If given figure, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and
Page 3 :
∠APB = 60°. Find the length of chord AB., , Solution:, , Question 6., In figure, 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., , Solution:, , Question 7., In given figure, from an external point P, two tangents PT and PS are drawn to a circle with, centre O and radius r.If PO = 2r, show that ∠OTS =∠OST = 30°.
Page 4 :
Solution:, , Question 8., In given figure, from a point P, two tangents PT and PS are drawn to a circle with centre O, such that ∠SPT = 120°, Prove that OP = 2PS, , Solution:, , Question 9., In given figure, there are two concentric circles of radii 6 cm and 4 cm with centre O. If AP is, a tangent to the larger circle and BP to the smaller circle and length of AP is 8cm, find the, length of BP, , Solution:
Page 5 :
Long Answer Type Questions [4 Marks], Question 10., Prove that the lengths of tangents drawn from an external point to a circle are equal, Solution:, , Question 11., , Solution:, , Question 12., Prove that the tangent at any point of a circle is perpendicular to the radius through the point, of contact, Solution:
Page 6 :
Question 13., In given figure, two equal circles, with centres O and O’, touch each other at X. OO’, produced meets the circle with centre O’ at A. AC is tangent to the circle with centre O, at, the point C. O’D is perpendicular to AC. Find the value of DO’/CO., , Solution:, , Question 14., In given figure, AB is a chord of a circle, with centre O, such that AB = 16 cm and radius of, circle is 10 cm. Tangents at A and B intersect each other at P. Find the length of PA, , Solution:
Page 7 :
2015, Very Short Answer Type Questions [1 Mark, Question 15., In figure, PA and PB are tangents to the circle with centre O such that ∠APB = 50°. Write, the measure of ∠OAB, Solution:, , Question 16., Find the relation between x and y if the points A(x, y), B(-5, 7) and C(-4, 5) are collinear, Solution:
Page 8 :
Question 17., Two concentric circles of radii a and b(a > b) are given. Find the length of the chord of the, larger circle which touches the smaller circle., Solution:, , Short Answer Type Questions I [2 Marks], Question 18., In figure, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°,, find ∠ATQ, , Solution:, , Question 19., From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle.
Page 9 :
Prove that OT is the right bisector of the line segment PQ., Solution:, , Question 20., In figure, two tangents RQ and RP are drawn from an external point R to the circle with, centre O. If ∠PRQ = 120°, then prove that OR = PR + RQ., , Solution:, , Question 21., In figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the, segments BD and DC are respectively of lengths 6 cm and 9 cm. If the area of ΔABC is 54, cm², then find the lengths of sides AB and AC
Page 10 :
Solution:, , Question 22., In figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external, point P. If ∠TPQ = 70°, find ∠TRQ, , Solution:
Page 11 :
Question 23., In figure, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q, intersect at a point T. Find the lengths of TP and TQ, , Solution:, , Long Answer Type Questions [4 Marks], Question 24., In figure, tangents PQ and PR are drawn from an external point P to a circle with centre O,, such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS, , Solution:
Page 12 :
Question 25., Prove that the tangent at any point of a circle is perpendicular to the radius throug the point, of contact, Solution:, Refer to Ans. 12, Question 26., Prove that the lengths of the tangents drawn from an external point to a circle are equal, Solution:, Refer to Ans. 10., Question 27., 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., Solution:
Page 13 :
Question 28., In figure, O is the centre of the circle and TP is the tangent to the circle from an external, point T. If ∠PBT = 30°, prove that BA: AT = 2:1, , Solution:, , 2014, Short Answer Type Questions I [2 Marks], Question 29., Prove that the line segment joining the points of contact of two parallel tangents of a circle,, passes through its centre.
Page 14 :
Solution:, , Question 30., If from an external point P of a circle with centre O, two tangents PQ and PR are drawn, such, that ∠QPR = 120°, prove that 2PQ = PO., Solution:, , Question 31., In figure, common tangents AB and CD to the two circles with Centres O1 and O2 intersect, at E. Prove that AB = CD., , Solution:
Page 15 :
Question 32., The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and, AB at D, E and F respectively. Prove that BD = DC, Solution:, , Question 33., In figure, XP and XQ are two tangents to the circle with centre O, drawn from an external, point X. ARB is another tangent, touching the circle at R. Prove that XA+AR=XB+BR., Solution:, , Question 34., Prove that the tangents drawn at the ends of any diameter of a circle are parallel., Solution:, , Long Answer Type Questions [4 Marks], Question 35., Prove that the length of the tangents drawn from an external point to a circle are equal., Solution:, Refer to Ans. 10., Question 36., Prove that a parallelogram circumscribing a circle is a rhombus, Solution:
Page 16 :
Question 37., Prove that the tangent at any point of a circle is perpendicular to the radius through the point, of contact, Solution:, Refer to Ans. 12., , Question 38., In figure, PQ is a chord of length 16 cm, of a circle of radius 10 cm. The tangents at P and Q, intersect at a point T. Find the length of TP., , Solution:
Page 17 :
Question 39., Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary, angles at the centre of the circle., Solution:
Page 18 :
Question 40., In figure, a triangle ABC is drawn to circumscribe a circle of radius 4 cm, such that the, segments BD and DC are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC., , Solution:
Page 19 :
Question 41., Prove that the lengths of the tangents drawn from an external point to a circle are equal., Solution:, , Question 42., A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are, equal, Solution:, , 2013, Short Answer Type Questions I [2 Marks], Question 43., Prove that the parallelogram circumscribing a circle is a rhombus, Solution:, Refer to Ans. 36., Question 44., In the given figure, a circle inscribed in ΔABC touches its sides AB, BC and AC at points D, E, and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then find the lengths of AD,, BE and CF
Page 20 :
Solution:, , Question 45., In the 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., , Solution:, , Question 46., In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB +, CD = AD + BC, Solution:
Page 21 :
Question 47., In the given figure, a circle inscribed in ΔABC, touches its sides BC, CA and AB at the points, P, Q and R respectively. If AB = AC, then prove that BP = CP., , Solution:, , Question 48., In the given figure, two tangents PA and PB are drawn to a circle with centre O from an, external point P. Prove that ∠APB = 2 ∠OAB, , Solution:, , Long Answer Type Questions [4 Marks], Question 49., Prove that the tangent at any point of a circle is perpendicular to the radius through the point, of contact, Solution:, Refer to ANS.12
Page 22 :
Question 50., In the given figure l,m are two parallel tangents to the circle with center O,touching the circle, at A and B respectively.Another tangent at C intersect the line l at D and m at E. prove that, ∠DOE=90, , Solution:, , Question 51., Prove that the lengths of tangents drawn from an external point to a circle are equal., Solution:, , Question 52., In the given figure, PA and PB are two tangents drawn from an external point P to a circle, with centre O. Prove that OP is the right bisector of line segment AB.
Page 23 :
Solution:, , Question 53., Prove that the tangent at any point of a circle is perpendicular to the radius through the point, of contact, Solution:, , Question 54., In the given figure, the sides AB, BC and CA of ΔABC touch a circle with centre O and, radius r at P, Q and R respectively., Prove that:, 1. AB + CQ = AC + BQ, 2. Area (ΔABC) = 1/2 (perimeter of ΔABC) X r
Page 24 :
Solution:, , 2012, Short Answer Type Questions I [2 Marks], Question 55., Tangents PA and PB are drawn from an external point P to two concentric circles with centre, O and radii 8 cm and 5 cm respectively, as shown in Fig. If AP = 15 cm, then find the length, of BP, , Solution:, , Question 56., In figure, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the, point of contact P bisects the base BC
Page 25 :
Solution:, , Question 57., In figure, the chord AB of the larger of the two concentric circles, with centre O, touches the, smaller circle at C. Prove that AC = CB., , Solution:, , Question 58., In figure, a right triangle ABC, circumscribes a circle of radius r. If AB and BC are of lengths, 8 cm and 6 cm respectively, find the value of r.
Page 26 :
Solution:, , Question 59., Prove that the tangents drawn at the ends of a diameter of a circle are parallel, Solution:, , Question 60., The incircle of an isosceles triangle ABC, with AB = AC, touches the sides AB, BC and CA at, D, E and F respectively. Prove that E bisects BC, Solution:, , Question 61., Prove that in two concentric circles, the chord of the larger circle, which touches the smaller, circle, is bisected at the point of contact, Solution:
Page 27 :
Short Answer Type Questions II [3 Marks], Question 62., Prove that the parallelogram circumscribing a circle is a rhombus., Solution:, , Question 63., Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary, angles at the centre of the circle., Solution:
Page 28 :
Question 64., In figure, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR = 12, cm. Find the lengths QM, RN and PL., , Solution:, , Question 65., Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove, that ∠PTQ = 2∠OPQ, Solution:
Page 29 :
Question 66., In figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent, AB with point of contact C intersects XY at A and X’Y’ at B. Prove that ∠AOB = 90°., , Solution:
Page 30 :
Long Answer Type Questions [4 Marks], Question 67., Prove that the tangent at any point of a circle is perpendicular to the radius through the point, of contact, Solution:, Refer to Ans. 12., Question 68., A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC., Solution:, Refer to Ans. 46., Question 69., Prove that the lengths of tangents drawn from an external point to a circle are equal. Using it,, prove: quadrilateral ABCD is drawn to circumscribe a circle. Such’that AB + CD = AD + BC, , Solution:, , Question 70., Prove that the tangent at any point of a circle is perpendicular to the radius through the point, of contact., Solution:
Page 31 :
Refer to Ans. 12., 2011, Short Answer Type Questions I [2 Marks], Question 71., Two concentric circles are of radii 7 cm and r cm respectively, where r >7 .A chord of the, larger circle, of length 48 cm, touches the smaller circle. Find the value of r., Solution:, , Question 72., In figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6, cm, BC = 9 cm and CD = 8 cm. Find the length of side AD., , Solution:, , Question 73., 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 + d²., Solution:
Page 32 :
Short Answer Type Questions II [3 Marks], Question 74., , Solution:
Page 33 :
Long Answer Type Questions [4 Marks], Question 75., Prove that the lengths of tangents drawn from an external point to a circle are equal., Solution:, , Question 76., Prove that the tangent at any point of a circle is perpendicular to the radius through the point, of contact, Solution:, Refer to Ans. 12., Question 77., Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary, angles at the centre of the circle., Solution:
Page 34 :
2010, Very Short Answer Type Questions [1 Mark], Question 78., In figure, CP and CQ are tangents from an external point C to a circle with centre O. AB is, another tangent which touches the circle at R. If CP = 11 cm and BR = 4 cm, find the length, of BC., Solution:, , Question 79., 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 = 13 cm. Find the length PQ., Solution:, , Short Answer Type Questions I [2 Marks], Question 80., Prove that the lengths of tangents drawn from an external point to a circle are equal. Using, the above prove the following: In Fig., PA and PB are tangents from an external point P , to a, circle with centre O. LN touches the circle at M. Prove that PL + LM = PN + MN., Solution:, Refer to Ans. 10 and 33.
Page 35 :
Question 81., In figure, there are two concentric circles, with centre O and of radii 5 cm and 3 cm. From an, external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find the, length of BP., , Solution:
