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10
NCERT Solutions · Maths Ch 10
Circles
20 questions · 2 exercises
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Ex 10.1
Ex 10.2
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
7 QUESTIONS
Exercise 10.1
Q1
How many tangents can a circle have?
(1 mark)
Q2(i)
Fill in the blank: A tangent to a circle intersects it in ______ point(s).
(1 mark)
Q2(ii)
Fill in the blank: A line intersecting a circle in two points is called a ______.
(1 mark)
Q2(iii)
Fill in the blank: A circle can have ______ parallel tangents at the most.
(1 mark)
Q2(iv)
Fill in the blank: The common point of a tangent to a circle and the circle is called ______.
(1 mark)
Q3
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 = 12 cm. Length PQ is: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm
(1 mark)
Q4
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
(2 marks)
2
13 QUESTIONS
Exercise 10.2
Q1
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm
(1 mark)
Q2
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to (A) 60° (B) 70° (C) 80° (D) 90°
Figure: a circle with centre O; tangents TP and TQ from an external point T touch the circle at P and Q. OP and OQ are joined.
(1 mark)
Q3
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to (A) 50° (B) 60° (C) 70° (D) 80°
(1 mark)
Q4
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
(3 marks)
Q5
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
(3 marks)
Q6
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
(2 marks)
Q7
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
(3 marks)
Q8
A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC.
Figure: a circle inside quadrilateral ABCD, touching AB at P, BC at Q, CD at R and DA at S.
(3 marks)
Q9
In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠AOB = 90°.
Figure: a circle with centre O between two parallel tangents XY (touching at P) and X′Y′ (touching at Q). A third tangent touches the circle at C and meets XY at A and X′Y′ at B.
(5 marks)
Q10
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.
(3 marks)
Q11
Prove that the parallelogram circumscribing a circle is a rhombus.
(3 marks)
Q12
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.
Figure: a circle with centre O and radius 4 cm inside △ABC, touching BC at D, CA at E and AB at F. BD = 8 cm, DC = 6 cm.
(5 marks)
Q13
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
(5 marks)