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.
Related Articles: