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Class 10 Maths Chapter 10: Circles MCQs with answers

10 multiple-choice questions on Circles, each with its answer and the reason, written from the NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter.

Question 1
A straight cycle track runs past a circular pond and shares exactly one point K with the pond's edge. With respect to the circle of the pond's edge, the track is
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Answer: (c) a tangent

A line that meets a circle in exactly one point is a tangent, and K is its point of contact.

Question 2
A point P is 6 cm from the centre of a circle of radius 6 cm. The number of tangents to the circle that pass through P is
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Answer: (b) 1

OP = 6 cm equals the radius, so P lies on the circle. At a point on the circle there is exactly one tangent.

Question 3
A line l is at a distance of 7 cm from the centre of a circle of radius 6.5 cm. Then l is
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Answer: (c) a line that does not meet the circle

The distance 7 cm is greater than the radius 6.5 cm, so every point of l lies outside the circle: l and the circle have no common point.

Question 4
Which of the following statements is true?
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Answer: (b) A circle has infinitely many tangents, but only two of them pass through a given point outside it.

There is one tangent at every point of the circle, so infinitely many in all; through a point outside the circle exactly two of them pass, and through a point inside, none.

Question 5
PT is a tangent at P to a circle with centre O, and T is a point on the tangent. If ∠TOP = 52°, then ∠OTP is
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Answer: (a) 38°

The tangent is perpendicular to the radius, so ∠OPT = 90°. In ΔOPT, ∠OTP = 180° − 90° − 52° = 38°.

Question 6
The length of the tangent from a point 17 cm from the centre of a circle of radius 8 cm is
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Answer: (a) 15 cm

The right angle is at the point of contact, so the tangent length is √(17² − 8²) = √225 = 15 cm.

Question 7
A straight lane from a tea stall K just touches a circular running track of radius 35 m, and the stretch of lane from K to the point of contact is 12 m long. The distance of K from the centre of the track is
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Answer: (b) 37 m

The radius to the point of contact is perpendicular to the lane, so the distance from K to the centre is the hypotenuse: √(35² + 12²) = √1369 = 37 m. (√(35² − 12²) treats the distance as a leg.)

Question 8
PA is a tangent from P to a circle of radius 12 cm with centre O. The line PO meets the circle first at C, and PC = 25 cm. The length of the tangent PA is
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Answer: (c) 35 cm

OP = PC + OC = 25 + 12 = 37 cm. The tangent length is PA = √(37² − 12²) = √1225 = 35 cm, measured to the point of contact, not to C or to the far side.

Question 9
PA and PB are the tangents from P to a circle. If PA = (3x + 1) cm and PB = (5x − 7) cm, then x equals
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Answer: (a) 4

Tangents from an external point are equal: 3x + 1 = 5x − 7, so 2x = 8 and x = 4.

Question 10
A chord PQ of a circle with centre O is equal in length to the radius. The tangents at P and Q meet at T. Then ∠PTQ is
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Answer: (b) 120°

OP = OQ = PQ, so ΔOPQ is equilateral and ∠POQ = 60°. ∠PTQ and ∠POQ are supplementary: ∠PTQ = 180° − 60° = 120°.

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