Class 10 - Chapter 10: Circles

Exercise 10.1

Q1. How many tangents can a circle have?

A circle has infinitely many points on its circumference.

A tangent can be drawn at each point of the circle.

Therefore, a circle can have infinitely many tangents.

Q2. Fill in the blanks:

(i) A tangent to a circle intersects it in ______ point(s).

A tangent touches the circle at exactly one point.

Answer: one

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

A line that cuts a circle at two distinct points is called a secant.

Answer: secant

(iii) A circle can have ______ parallel tangents at the most.

At most, two tangents to a circle can be parallel.

Answer: two

(iv) The common point of a tangent to a circle and the circle is called ______.

The point where the tangent touches the circle is called the point of contact.

Answer: point of contact

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 cm. Find PQ.

Given:

OP = 5 cm (radius)

OQ = 12 cm

PQ is a tangent at P.

Radius drawn to the point of contact is perpendicular to the tangent.

Therefore,

∠OPQ = 90°

In right △OPQ,

OQ² = OP² + PQ²

12² = 5² + PQ²

144 = 25 + PQ²

PQ² = 119

PQ = √119 cm

Answer: √119 cm (Option D)

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.

Construction Steps:

1. Draw a circle with centre O.
2. Draw any straight line l.
3. Draw a line m parallel to l such that it touches the circle at exactly one point. Line m is the tangent.
4. Draw another line n parallel to l and m such that it cuts the circle at two points. Line n is the secant.

Thus, m is a tangent and n is a secant, both parallel to the given line l.