Class 10 Mathematics — Circles

MCQs + Fill in the Blanks + True/False + Assertion–Reason + Very Short Answer

A. MCQs

1. A line which intersects a circle at exactly one point is called:
(a) Secant (b) Tangent (c) Chord (d) Normal

2. A line intersecting a circle at two distinct points is called:
(a) Tangent (b) Radius (c) Secant (d) Normal

3. The angle between a tangent and the radius drawn to the point of contact is:
(a) $45^\circ$ (b) $60^\circ$ (c) $90^\circ$ (d) $180^\circ$

4. From a point inside a circle, the number of tangents that can be drawn is:
(a) 0 (b) 1 (c) 2 (d) Infinitely many

5. From a point on a circle, the number of tangents that can be drawn is:
(a) 0 (b) 1 (c) 2 (d) 3

6. From an external point, the number of tangents that can be drawn to a circle is:
(a) 1 (b) 2 (c) 3 (d) 0

7. If $PT$ and $PQ$ are tangents drawn from the same external point $P$, then:
(a) $PT>PQ$ (b) $PT<PQ$ (c) $PT=PQ$ (d) Cannot be determined

8. If the radius of a circle is $5$ cm and the distance of an external point from its centre is $13$ cm, the tangent length is:
(a) 8 cm (b) 10 cm (c) 12 cm (d) 18 cm

9. If two tangents from an external point make an angle of $70^\circ$, the angle subtended by the line segment joining their points of contact at the centre is:
(a) $70^\circ$ (b) $90^\circ$ (c) $110^\circ$ (d) $140^\circ$

10. If a tangent touches a circle at $P$, which statement is always true?
(a) $OP\parallel$ tangent
(b) $OP=$ tangent
(c) $OP\perp$ tangent
(d) $OP$ bisects tangent

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B. Fill in the Blanks

11. A tangent intersects a circle at ______ point.

12. A secant intersects a circle at ______ points.

13. The common point of a tangent and a circle is called the point of ______.

14. The tangent at any point of a circle is ______ to the radius through that point.

15. From a point inside a circle, ______ tangent can be drawn.

16. From a point outside a circle, ______ tangents can be drawn.

17. Tangents drawn from the same external point to a circle are ______ in length.

18. The perpendicular from the centre of a circle to a chord ______ the chord.

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C. True or False

19. A tangent can intersect a circle at two distinct points.

20. There is exactly one tangent at a particular point of a circle.

21. Two tangents drawn from the same external point are equal in length.

22. A radius through the point of contact is parallel to the tangent.

23. A point inside a circle has no tangent to the circle passing through it.

24. The centre lies on the angle bisector of the angle between two tangents drawn from an external point.

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D. Assertion–Reason

Choose:
(a) Both A and R are true, and R correctly explains A.
(b) Both A and R are true, but R does not correctly explain A.
(c) A is true, but R is false.
(d) A is false, but R is true.

25.
A: The tangent at a point of a circle is perpendicular to the radius through that point.
R: The radius gives the shortest distance from the centre to the tangent line.

26.
A: Two tangents drawn from an external point have equal lengths.
R: The two right triangles formed using the centre, external point and points of contact are congruent by RHS.

27.
A: Two tangents can always be drawn from a point inside a circle.
R: Every line through an interior point intersects the circle at two points.

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E. Very Short Answer

28. What is a tangent to a circle?

29. What is a secant?

30. How many tangents can be drawn from a point lying outside a circle?

31. If $OP=10$ cm and the radius is $6$ cm, find the tangent length from $P$.

32. Two tangents from an external point have lengths $7.5$ cm and $x$ cm. Find $x$.

33. A tangent touches a circle at $P$. What is $\angle OPT$, where $O$ is the centre and $PT$ is the tangent?

34. If the angle between two tangents is $80^\circ$, find the angle subtended by the segment joining the points of contact at the centre.

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Answers

MCQs

1. (b)
2. (c)
3. (c)
4. (a)
5. (b)
6. (b)
7. (c)
8. (c) — $PT=\sqrt{13^2-5^2}=12$ cm
9. (c) — $180^\circ-70^\circ=110^\circ$
10. (c)

Fill in the Blanks

11. one
12. two
13. contact
14. perpendicular
15. no
16. two
17. equal
18. bisects

True/False

19. False
20. True
21. True
22. False
23. True
24. True

Assertion–Reason

25. (a)
26. (a)
27. (d)

Very Short Answer

28. A line that meets a circle at exactly one point.
29. A line that intersects a circle at two points.
30. Two.
31. $PT=\sqrt{10^2-6^2}=\boxed{8\text{ cm}}$
32. $\boxed{7.5\text{ cm}}$
33. $\boxed{90^\circ}$
34. $\boxed{100^\circ}$