NCERT Class 10 Maths – Exercise 10.1 Solutions

NCERT Class 10 Maths

Chapter 10 – Circles | Exercise 10.1

(Rationalized Syllabus 2025-26)

💡 Key Concepts

  • Tangent to a Circle: A line that intersects the circle at exactly one point.
  • Secant: A line that intersects the circle at two distinct points.
  • Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Q1

How many tangents can a circle have?

A circle is a collection of an infinite number of points.
At every point on the circle, a unique tangent can be drawn.
✔️ Infinitely many
Q2

Fill in the blanks:

(i) A tangent to a circle intersects it in one point(s).
(ii) A line intersecting a circle in two points is called a secant.
(iii) A circle can have two parallel tangents at the most. (These occur at the ends of a diameter).
(iv) The common point of a tangent to a circle and the circle is called 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. Length PQ is: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) $\sqrt{119}$ cm

Geometry Setup
We have a circle with center $O$. Radius $OP = 5$ cm.
$PQ$ is the tangent at point $P$.
Line connecting Center $O$ to external point $Q$ has length $OQ = 12$ cm.
Calculation
According to Theorem 10.1, the radius is perpendicular to the tangent at the point of contact. Therefore, $\angle OPQ = 90^\circ$.
In right-angled $\triangle OPQ$, using Pythagoras theorem:
$$OP^2 + PQ^2 = OQ^2$$
$$5^2 + PQ^2 = 12^2$$
$$25 + PQ^2 = 144$$
$$PQ^2 = 144 – 25 = 119$$
$$PQ = \sqrt{119} \text{ cm}$$
✔️ (D) $\sqrt{119}$ cm
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.

[Image of Circle with parallel tangent and secant]
Steps of Construction:
  1. Draw a circle with any radius and center O.
  2. Draw a straight line $XY$ anywhere near the circle (this is the given line).
  3. Draw a radius perpendicular to the given line $XY$.
  4. At the point where the radius meets the circle (say point P), draw a line perpendicular to the radius. This line will be the tangent and will be parallel to $XY$.
  5. Draw another line strictly inside the circle perpendicular to the same radius. This line will intersect the circle at two points and acts as the secant parallel to $XY$.
🎉 Exercise 10.1 Completed | Chapter 10 Circles
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