NCERT Solutions for class 10th Maths Chapter 10 Circles

Exercise 10.1

Question 1. How many tangents can a circle have?
Sol. There can be infinite number of tangents to a circle.

Question 2. Fill in the blanks :
(I) A tangent to a circle intersects it in _ point (s).
(ii) A line intersecting a circle in two points is called a
(iii) A circle can have_____parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called _

Sol. (i) One (ii) Secant (iii) Two (iv) Point of contact.

Question 3. 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 :

Sol. (d) Given O is the centre of the circle. The radius of the circle is 5 cm. And PQ is tangent to the circle at P.

Question 4. Draw a circle and two lines parallel to a given line such that one is a tangent and other a secant to the circle.

1. Consider a circle with centre 0. Draw a line PQ, outside the circle.
2. With centre 0, draw an arc cutting the line PQ at A& B.
3. With centre A & B respectively radius more than 1/2 AB draw two arc’s
above the line PQ, cutting each other at D. Join O and D.
4. Line OD meets the circle at y and draw lines RS through y, perpendicular to OD.
5. Now, consider a point Z on OD inside the circle. Through Z draw line TU perpendicular to OD.
RS and TU are the required lines.

Exercise 10.2

In Q.1 to 3, choose the correct option and give justification.

Question 1. 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

Sol (a) In AOQ, by pythagoras theorem: AQ2 = A02 + Q02 From figure,r2 = (25)2-(24)2 = 625 – 576 = 49 r = 7 cm

Question 2. In Fig. if TP and TQ are the two tangents to a circle with centre O so that LPOQ = 110°, then LPTQ is equal to

Question 3. 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 LPOA is equal to
(a) 50° (b) 60° (c) 70° (d) 80°

Question 4. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Sol. In figure AB is a diameter of the circle having centre at 0. l and m are two tangents drawn to the circle at A and B respectively.

Question 5. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Question 6. The length of a tangent from a point A at a distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Question 7. 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.
Hint : Concentric circles are those circles that have same centre.

Question 8. A quadrilateral ABCD is drawn to circumscribe a circle, Prove that AB+CD=AD+BC

Note: For all quadrilaterals circumscribing circle, sum of opposite pairs of sides are equal

Question 9. PQ and RS are two parallel tangents to a circle with centre O and another tangent XY with point of contact C intersect PQ at A and RS at B. Prove that LAOB = 90°.

Question 10. 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.

Sol. In the figure, PA and PB are the two tangents drawn to the cricle from a point P outside the circle. 0 is the centre of the circle. A and B are the points of contact of the tangents. AB is the line segment joining A and B. LAOB is subtended by the segment AB at the centre 0.

Note: Sum of opposite angles of a cyclic quadrilateral is 180°.
Question 11. Prove that the parallelogram circumscribing a circle is a rhombus.

Question 12. The radius of the incircle of a triangle is 4 cm and the secant into which the side is divided by the point of contact are 6 cm, 8 cm. Find the other two sides of the triangle.

Question 13. A circle touches all four sides of a quadrilateral ABCD. Prove that the angles subtended at the centre of the circle by the opposite sides are supplementary.

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