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 ............

(i) one

(ii) secant

(iii) two

(iv) point of contact

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

The line drawn from the centre of the circle to the tangent is perpendicular to the tangent.

∴ OP ⊥ PQ

By Pythagoras theorem in ΔOPQ,

OQ^{2} = OP^{2} +^{ }PQ^{2}

⇒ (12)^{2 }= 5^{2} + PQ^{2}

⇒PQ^{2} = 144 - 25

⇒PQ^{2} = 119

⇒PQ = √119 cm

(D) is the correct option.

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

AB and XY are two parallel lines where AB is the tangent to the circle at point C while XY is the secant to the circle.

choose the correct option and give justification.

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

The line drawn from the centre of the circle to the tangent is perpendicular to the tangent.

∴ OP ⊥ PQ

also, ΔOPQ is right angled.

OQ = 25 cm and PQ = 24 cm (Given)

By Pythagoras theorem in ΔOPQ,

OQ^{2} = OP^{2} +^{ }PQ^{2}

⇒ (25)^{2 }= OP^{2} + (24)^{2}

⇒ OP^{2} = 625 - 576

⇒ OP^{2} = 49

⇒ OP = 7 cm

The radius of the circle is option (A) 7 cm.