How many tangents can a circle have?

Fill in the blanks:

(1) A tangent to a circle intersects it in _______ point (s).

(2) A line intersecting a circle in two points is called a ______.

(3) A circle can have _______ parallel tangents at the most.

(4) The common point of a tangent to a circle and the circle is called ________.

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.

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.

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

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) 60⁰                      

(B) 70⁰                    

(C) 80⁰                       

(D) 90⁰

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) 50⁰                       

(B) 60⁰                             

(C) 70⁰                          

(D) 80⁰

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

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

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.

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.

A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that: AB + CD = AD + BC

 

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 = 90⁰.

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.

Prove that the parallelogram circumscribing a circle is a rhombus.

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.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.