Circles - Exercise 10.1

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Overview

Step-by-step NCERT solutions and practice problems for CBSE Class 10 Circles (Exercise 10.1), explaining tangent definitions, secant properties, and Theorem 10.1 (radius is perpendicular to tangent at point of contact).

Tangents, Secants, and Point of Contact Properties

Back to Chapters
Core Theorems Q1: Infinite Tangents Q2: Fill Blanks Q3: Tangent Length PQ Q4: Parallel Lines Construction
Key Theorems & Concepts:

Tangent: A straight line that touches 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 ($OP perp PQ$).

Question 1 NCERT Ex 10.1
How many tangents can a circle have?
Geometric Reasoning
A circle consists of an infinite collection of points lying on its circumference.
At each distinct point on the circumference, exactly one tangent line can be drawn.
Therefore, a circle can have infinitely many tangents.
A circle can have infinitely many tangents.
Question 2 NCERT Ex 10.1
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 _______ .
Explanations for Each Part
(i) one: By definition, a tangent is a line that intersects the circle at exactly one point.
(ii) secant: A straight line passing through a circle and cutting it at two points is called a secant.
(iii) two: Tangents drawn to a circle can be parallel only when they are drawn at the opposite endpoints of a diameter. Hence, there can be at most two parallel tangents in a given direction.
(iv) point of contact: The unique single point where the tangent touches the circle is known as the point of contact.
(i) one, (ii) secant, (iii) two, (iv) point of contact.
Question 3 NCERT Ex 10.1
A tangent $PQ$ at a point $P$ of a circle of radius $5 ext{ cm}$ meets a line through the centre $O$ at a point $Q$ so that $OQ = 12 ext{ cm}$. Length $PQ$ is:

(A) $12 ext{ cm}$     (B) $13 ext{ cm}$     (C) $8.5 ext{ cm}$     (D) $sqrt{119} ext{ cm}$
O P Q 5 cm PQ = ? 12 cm
Step 1: Apply Theorem 10.1
By Theorem 10.1, the radius through the point of contact is perpendicular to the tangent: $$OP perp PQ implies angle OPQ = 90^circ$$ Thus, $Delta OPQ$ is a right-angled triangle with hypotenuse $OQ = 12 ext{ cm}$ and radius $OP = 5 ext{ cm}$.
Step 2: Use Pythagoras Theorem
$$OQ^2 = OP^2 + PQ^2$$ $$12^2 = 5^2 + PQ^2$$ $$144 = 25 + PQ^2$$ $$PQ^2 = 144 - 25 = 119$$ $$PQ = sqrt{119} ext{ cm}$$
Length $PQ = sqrt{119} ext{ cm}$. Option (D) is correct.
Question 4 NCERT Ex 10.1
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.
O Given Line $l$ Secant $m$ Tangent $n$
Step-by-Step Construction Procedure
1. Draw a circle with centre $O$ and any convenient radius.
2. Draw a straight line $l$ outside the circle (the given line).
3. Draw a line $m$ parallel to line $l$ intersecting the circle at two distinct points. Line $m$ is a secant.
4. Draw a line $n$ parallel to line $l$ touching the circle at exactly one point (point of contact). Line $n$ is a tangent.
Here, line $m parallel ext{line } n parallel ext{line } l$, satisfying the required condition.
Construction completed: Secant $m$ and tangent $n$ are both parallel to given line $l$.
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