Chapter 10: Circles

VVI questions — the ones the board asked most (17)

1. Q1Long answer5 marksAsked: 2022

   Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.

   

   Medha Ma’am

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   Model answer · tangents, diameter proof

   1. 1.We are given a circle with centre O and two parallel tangents AB and CD drawn to the circle, touching the circle at points P and Q respectively.1 mark
   2. 2.Since OP is the radius through the point of contact P of tangent AB, by the theorem that the tangent at any point of a circle is perpendicular to the radius through the point of contact, OP is perpendicular to AB. Similarly, since OQ is the radius through the point of contact Q of tangent CD, OQ is perpendicular to CD.1 mark
   3. 3.Since AB is parallel to CD, and OP is perpendicular to AB while OQ is perpendicular to CD, both OP and OQ are perpendicular to the same pair of parallel lines. A line perpendicular to one of two parallel lines is also perpendicular to the other, and through a point only one line can be drawn perpendicular to a given line. Therefore, OP and OQ must lie along the same straight line, which means the points P, O and Q are collinear.1.5 marks
   4. 4.Since P, O and Q are collinear, the line segment PQ passes through the centre O of the circle.0.5 marks
   5. 5.Since the line segment PQ joins two points P and Q on the circle and passes through the centre O, PQ is the diameter of the circle. Hence, it is proved that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.1 mark

   Diagram: Draw a circle with centre O; draw two parallel tangents AB and CD touching the circle at points P and Q respectively, with AB on one side and CD on the opposite side; join OP, OQ and show the line segment PQ passing through O; label O, P, Q, A, B, C, D.

   • Clearly state that OP ⟂ AB and OQ ⟂ CD using the tangent-radius perpendicularity fact before concluding collinearity.
   • Explicitly write that P, O, Q are collinear because both OP and OQ are perpendicular to the parallel tangents.
   • End with the sentence 'PQ is the diameter of the circle' to earn the final proved point.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

2. Q2Long answer5 marksAsked: 2022

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

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 105 s

   Model answer · tangents at ends of diameter, parallel

   1. 1.Let O be the centre of a circle and AB be a diameter of the circle. Let PQ be the tangent drawn at the point A and RS be the tangent drawn at the point B. We need to prove that PQ is parallel to RS.1 mark
   2. 2.The tangent at any point of a circle is perpendicular to the radius through the point of contact. Since OA is the radius through the point of contact A, and PQ is the tangent at A, OA is perpendicular to PQ. Similarly, since OB is the radius through the point of contact B, and RS is the tangent at B, OB is perpendicular to RS.1.5 marks
   3. 3.Since OA is perpendicular to PQ, the angle OAP is equal to 90°. Since OB is perpendicular to RS, the angle OBS is equal to 90°.1 mark
   4. 4.Since AB is a straight line (the diameter) cutting the two lines PQ and RS, and the angle OAP and the angle OBS are formed on opposite sides of the line AB at the points A and B, these two angles are alternate interior angles. We have shown that angle OAP = 90° and angle OBS = 90°, so angle OAP = angle OBS. Since a transversal AB makes equal alternate interior angles with the lines PQ and RS, the line PQ is parallel to the line RS. Hence, the tangents drawn at the ends of a diameter of a circle are parallel.1.5 marks

   Diagram: Draw a circle with centre O and a diameter AB; draw tangent line PQ touching the circle at A (perpendicular to OA) and tangent line RS touching the circle at B (perpendicular to OB), labelling O, A, B, P, Q, R, S and marking the right angles ∠OAP and ∠OBS.

   • State clearly that O is the centre and name the tangents PQ at A and RS at B.
   • Write the reason 'tangent is perpendicular to radius through point of contact' before concluding OA ⊥ PQ and OB ⊥ RS.
   • Explicitly write angle OAP = 90° and angle OBS = 90°, then call them alternate interior angles equal to conclude PQ is parallel to RS.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

3. Q3Long answer5 marksAsked: 2023

   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.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 130 s

   Model answer · tangent angle, circle theorem

   1. 1.We are given a circle with centre O and an external point P from which two tangents PA and PB are drawn to the circle, touching it at points A and B. We join OA and OB. Since the tangent to a circle is perpendicular to the radius through the point of contact, OA is perpendicular to PA, so ∠OAP = 90°. Similarly, OB is perpendicular to PB, so ∠OBP = 90°.1 mark
   2. 2.In the quadrilateral OAPB, O is the centre, A and B are the points of contact, and P is the external point. Since OAPB is a quadrilateral, the sum of its interior angles is 360°.1 mark
   3. 3.Taking the four angles of the quadrilateral OAPB in order, ∠OAP + ∠OBP + ∠APB + ∠AOB = 360°, where ∠APB is the angle between the two tangents at P and ∠AOB is the angle subtended at the centre O by the line segment AB joining the points of contact.1 mark
   4. 4.Substituting ∠OAP = 90° and ∠OBP = 90° in the above equation, we get 90° + 90° + ∠APB + ∠AOB = 360°.1 mark
   5. 5.This gives ∠APB + ∠AOB = 360° − 180° = 180°. Since ∠APB and ∠AOB add up to 180°, they are supplementary. Hence, 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. Hence proved.1 mark

   Diagram: Draw a circle with centre O, an external point P, two tangents PA and PB touching the circle at A and B; join OA, OB and AB; label O, P, A, B, and mark ∠OAP, ∠OBP, ∠APB and ∠AOB.

   • Write ∠OAP = 90° and ∠OBP = 90° with the reason 'tangent to a circle is perpendicular to the radius through the point of contact'.
   • Write the quadrilateral angle sum as its own line: ∠OAP + ∠OBP + ∠APB + ∠AOB = 360°.
   • Show the substitution 90° + 90° + ∠APB + ∠AOB = 360° leading to ∠APB + ∠AOB = 180°, and end with 'Hence proved'.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

4. Q4Long answer5 marksAsked: 2023

   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.

   How the model answer begins

   1. We are given a circle with centre O and an external point P from which two tangents PA and PB are drawn to the circle, touching it at points A and B. Since the tangent to a circle is perpendicular to the radius through…

   The full answer from Medha Ma’am →
5. Q5Long answer5 marksAsked: 2021

   If two circles are concentric, then prove that the chord of the larger circle which is tangent to the smaller circle is bisected at the point of contact.

   How the model answer begins

   1. We are given two concentric circles with centre O. AB is a chord of the larger circle which touches the smaller circle at the point P. We need to prove that AP = BP.

   The full answer from Medha Ma’am →
6. Q6Long answer5 marksAsked: 2021

   Draw a circle of radius 3 cm. From a point situated at a distance of 5 cm from the centre of the circle, draw two tangents to the circle and measure their lengths.

   How the model answer begins

   1. A circle of radius 3 cm is drawn with centre O.

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7. Q7Short answer2 marksAsked: 2026

   A circle is inscribed in a triangle ABC touching the sides BC, CA and AB of the triangle at P, Q and R respectively. If AB = 10 cm, AQ = 7cm and CQ = 5 cm then find the length of BC.

   How the model answer begins

   1. The lengths of tangents drawn from an external point to a circle are equal. Since A is an external point with tangents AR and AQ touching the circle at R and Q, AR = AQ = 7 cm. Since AB = 10 cm and AR = 7 cm, BR = AB -…

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8. Q8Short answer2 marksAsked: 2026

   If the lengths of a chord and radius of a circle are 24 cm and 13 cm respectively then find the perpendicular distance of the chord from the centre of the circle.

   How the model answer begins

   1. The perpendicular from the centre of a circle to a chord bisects the chord. Here the chord AB has length 24 cm, so the perpendicular from the centre O to the chord meets it at point M such that AM = MB = 12 cm.

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9. Q9Short answer2 marksAsked: 2026

   The chord QR of a circle makes an angle 130° at the centre O of the circle. The tangents at Q and R meet at the point P. Find the value of angle QPR.

   How the model answer begins

   1. The angle between the tangent and the radius is 90°. Here Q and R are the points of contact of the tangents from the external point P, and O is the centre of the circle. Since OQ is the radius and PQ is the tangent at…

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10. Q10Short answer2 marksAsked: 2022

    If PA and PB are two tangents to a circle with centre at O and angle APB = 60 degree, then find the measure of angle OAB.

    How the model answer begins

    1. In triangle OAP, OA is perpendicular to AP, so angle OAP = 90°, since the tangent to a circle is perpendicular to the radius through the point of contact.

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11. Q11Short answer2 marksAsked: 2022

    What is the length of a tangent drawn from a point Q, which is at a distance of 13 cm from the centre O of a circle, if radius of the circle is 5 cm ?

    How the model answer begins

    1. Let O be the centre of the circle, P the point of contact and Q the external point from which the tangent is drawn, with OQ = 13 cm and OP = 5 cm (radius). The tangent to a circle is perpendicular to the radius through…

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12. Q12Short answer2 marksAsked: 2023

    The length of the tangent drawn from an external point P to a circle is 15 cm and the distance of the point P from the centre of the circle is 17 cm. Find the radius of the circle.

    How the model answer begins

    1. The tangent to a circle is perpendicular to the radius through the point of contact. So if O is the centre of the circle, P is the external point and T is the point of contact, then OT is the radius, PT is the tangent,…

    The full answer from Medha Ma’am →
13. Q13Short answer2 marksAsked: 2023

    Prove that one and only one tangent can be drawn to any point on the circumference of a circle.

    How the model answer begins

    1. Let P be a point on the circumference of a circle with centre O, and let XY be the tangent to the circle at P. The tangent drawn at any point on the circle is perpendicular to the radius through the point of contact, so…

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14. Q14Short answer2 marksAsked: 2023

    Prove that one and only one tangent can be drawn to any point on the circumference of a circle

    How the model answer begins

    1. The tangent drawn at any point on the circle is perpendicular to the radius through the point of contact. Let O be the centre of the circle and P a point on the circle where the tangent XY touches it. OP is the shortest…

    The full answer from Medha Ma’am →
15. Q15Short answer2 marksAsked: 2023

    The length of the tangent drawn from an external point P to a circle is 15 cm and the distance of the point P from the centre of the circle is 17 cm. Find the radius of the circle.

    How the model answer begins

    1. The tangent to a circle is perpendicular to the radius through the point of contact. So the triangle OTP formed by the centre O, the point of contact T and the external point P is a right-angled triangle, right-angled…

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16. Q16Short answer2 marksAsked: 2021

    A point A is at a distance of 5 cm from the center O of a circle. If the length of the tangent AT to the circle is 3 cm, then what will be the length of the radius OT of the circle?

    How the model answer begins

    1. The tangent to a circle is perpendicular to the radius through the point of contact. Since AT is a tangent to the circle at point T and OT is the radius through the point of contact, OT is perpendicular to AT, so ∠OTA =…

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17. Q17Short answer2 marksAsked: 2021

    The radii of two concentric circles are 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

    How the model answer begins

    1. Let the two concentric circles have centre O, with the bigger circle of radius 5 cm and the smaller circle of radius 3 cm. Let AB be a chord of the bigger circle which touches the smaller circle at the point P. Since AB…

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Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   How many tangents can be drawn to a circle from an external point ?

   • (A) One
   • (B) Two ✓
   • (C) Infinite
   • (D) None of these

   Answer: (B)

2. 2. Bihar Board 2026

   If the lengths of radius of a circle and the tangent drawn from an external point to the circle are 7 cm and 24 cm respectively then the distance of the external point from the centre of the circle is

   • (A) 18 cm
   • (B) 32 cm
   • (C) 25 cm ✓
   • (D) 12 cm

   Answer: (C)

3. 3. Bihar Board 2026

   If the lengths of chord and radius of a circle are 16 cm and 17 cm respectively then perpendicular distance of the chord from the centre is

   • (A) 8 cm
   • (B) 15 cm ✓
   • (C) 14 cm
   • (D) 13 cm

   Answer: (B)

4. 4. Bihar Board 2026

   The sum ( in right angle ) of pair of opposite angles of a cyclic quadrilateral is

   • (A) 1/2
   • (B) 2/3
   • (C) 3/4
   • (D) 2 ✓

   Answer: (D)

5. 5. Bihar Board 2026

   Which of the following pairs of lines in a circle cannot be parallel?

   • (A) Two chords
   • (B) A chord and a tangent
   • (C) Two tangents
   • (D) Two diameters ✓

   Answer: (D)

6. 6. Bihar Board 2026

   If the angle between any two radii of a circle is 130° then the angle between the tangents at the ends of the radii is

   • (A) 65°
   • (B) 40°
   • (C) 50° ✓
   • (D) 90°

   Answer: (C)

7. 7. Bihar Board 2026

   The angle subtended at the centre of a circle is ∠AOB and ∠ACB where C lies on major arc at the circumference. Then ∠ACB =

   • (A) ∠AOB
   • (B) 2∠AOB
   • (C) ∠AOB/2 ✓
   • (D) None of these

   Answer: (C)

8. 8. Bihar Board 2026

   The length of the tangent PT, drawn from an external point P to a circle of radius 7 cm is 24 cm. If O is the centre of the circle, then the length of OP is

   • (A) 28 cm
   • (B) 18 cm
   • (C) 25 cm ✓
   • (D) 30 cm

   Answer: (C)

9. 9. Bihar Board 2026

   If the length of any chord of a circle is equal to the radius of the circle then the angle subtended by the chord at the centre is

   • (A) 120°
   • (B) 90°
   • (C) 60° ✓
   • (D) 30°

   Answer: (C)

10. 10. Bihar Board 2026

    If two circles touch each other externally then what is the number of common tangents?

    • (A) 1
    • (B) 2
    • (C) 3 ✓
    • (D) 4

    Answer: (C)

Questions parents and students ask

How many marks does "Circles" carry in Bihar Board Class 10 Maths?

In the Bihar Board papers of 2021–2026, this chapter carried about 4 marks and about 6 questions a year; by marks it ranks 12 in Maths.

Which are the most important (VVI) questions of "Circles"?

The 17 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 17 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.