NCERT Solutions for Class 10 Maths Chapter 10 Circles Exercise 10.1#

Exercise 10.1 introduces the concept of a tangent to a circle and the number of tangents from a point to a circle. It establishes the fundamental property that a tangent is perpendicular to the radius at the point of tangency.

Key Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

  • From a point inside the circle: 0 tangents
  • From a point on the circle: 1 tangent (at that point)
  • From a point outside the circle: 2 tangents

1. How many tangents can a circle have? #

Answer

A circle can have infinitely many tangents.

At every point on the circle, we can draw one tangent line (perpendicular to the radius at that point). Since a circle has infinitely many points, it has infinitely many tangents.

2. Fill in the blanks: #

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

A tangent to a circle intersects it in one point (the point of tangency/contact).

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

A line intersecting a circle in two points is called a secant.

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

A circle can have two parallel tangents at the most (one at each end of any diameter).

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

The common point of a tangent to a circle and the circle is called the point of contact (or point of tangency).

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: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) $\sqrt{119}$ cm #

Answer

PQ is the tangent at P and OP is the radius, so OP ⊥ PQ.

In right △OPQ, by Pythagoras Theorem:

$$PQ^2 = OQ^2 - OP^2$$

$$\Rightarrow PQ^2 = 12^2 - 5^2$$

$$\Rightarrow PQ^2 = 119$$

$$\Rightarrow PQ = \sqrt{119}$$

∴ Answer: (D) $\sqrt{119}$ cm

4. 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. #

Answer

Steps of construction:

  1. Draw a circle with centre O and a line $l$.
  2. Draw the diameter AB of the circle perpendicular to $l$.
  3. At A, draw a line perpendicular to OA. It is perpendicular to AB, so it is parallel to $l$. It touches the circle only at A, so it is a tangent.
  4. Through any point of AB between A and B, draw another line perpendicular to AB. It is also parallel to $l$, and it cuts the circle at two points, so it is a secant.

∴ The two lines drawn in steps 3 and 4 are both parallel to $l$. One is a tangent and the other is a secant.

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