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Mathematics · Textbook solutions

Circles

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 23 questions

10.2 Tangent to a Circle

7 q

Exercise 10.1

Practice · 7
  1. Ex 10.1 Q1
    How many tangents can a circle have?
  2. Fill in the blanks :
    Ex 10.1 Q2 (i)
    A tangent to a circle intersects it in ______ point(s).
  3. Ex 10.1 Q2 (ii)
    A line intersecting a circle in two points is called a ______.
  4. Ex 10.1 Q2 (iii)
    A circle can have ______ parallel tangents at the most.
  5. Ex 10.1 Q2 (iv)
    The common point of a tangent to a circle and the circle is called ______.
  6. Ex 10.1 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= 12 cm. Length PQ is :
    1. A.
      12 cm
    2. B.
      13 cm
    3. C.
      8.5 cm
    4. D.
      119\sqrt{119} cm
  7. Ex 10.1 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.

10.3 Number of Tangents from a Point on a Circle

16 q

Solved Examples

Worked · 3
  1. 10.2 Eg.1
    Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
  2. 10.2 Eg.2
    Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠\anglePTQ =2 ∠= 2\,\angleOPQ.
  3. 10.2 Eg.3
    PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Fig. 10.10). Find the length TP.

Exercise 10.2

Practice · 13
  1. In Q.1 to 3, choose the correct option and give justification.
    Ex 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
    1. A.
      7 cm
    2. B.
      12 cm
    3. C.
      15 cm
    4. D.
      24.5 cm
  2. Ex 10.2 Q2
    In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠\anglePOQ =110∘= 110^\circ, then ∠\anglePTQ is equal to
    1. A.
      60∘60^\circ
    2. B.
      70∘70^\circ
    3. C.
      80∘80^\circ
    4. D.
      90∘90^\circ
  3. Ex 10.2 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∘80^\circ, then ∠\anglePOA is equal to
    1. A.
      50∘50^\circ
    2. B.
      60∘60^\circ
    3. C.
      70∘70^\circ
    4. D.
      80∘80^\circ
  4. Ex 10.2 Q4
    Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
  5. Ex 10.2 Q5
    Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
  6. Ex 10.2 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.
  7. Ex 10.2 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.
  8. Ex 10.2 Q8
    A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB ++ CD == AD ++ BC
  9. Ex 10.2 Q9
    In Fig. 10.13, XY and X′^\primeY′^\prime 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′^\primeY′^\prime at B. Prove that ∠\angleAOB =90∘= 90^\circ.
  10. Ex 10.2 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.
  11. Ex 10.2 Q11
    Prove that the parallelogram circumscribing a circle is a rhombus.
  12. Ex 10.2 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.
  13. Ex 10.2 Q13
    Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.