### 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: AQ^{2} = A0^{2} + Q0^{2} From figure,r^{2} = (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 = 110Â°, 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 80Â°, then LPOA is equal to(a) 50Â° (b) 60Â° (c) 70Â° (d) 80Â°**

**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 = 90Â°.**

**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 lineÂ 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 180Â°.**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.**

**Related Articles:**

- NCERT Solutions for Class 10
- NCERT Solutions for Class 10 Maths
- Circles Class 10 Notes Maths Chapter 10