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Q1:
How many tangents can a circle have?
Solution :
Given: A circle in a two-dimensional Euclidean plane.
To Find: The total number of tangents that can be drawn to a circle.
Step 1: Defining a Tangent
A tangent to a circle is defined as a line that intersects the circle at exactly one point. This point is known as the point of contact. [Definition: A tangent is a line that touches the circle at a single point and does not enter the interior of the circle.]
Step 2: Analyzing the Circumference
A circle is defined as the locus of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center). The circumference of a circle consists of an infinite number of distinct points.
Step 3: Establishing the Correspondence
Since every point on the circumference of the circle can serve as a point of contact for a unique tangent line, we must determine the number of points on the circumference. [Axiom: A circle is composed of an infinite set of points.]
Step 4: Logical Deduction
1. Let $P$ be any arbitrary point on the circumference of the circle.
2. Through point $P$, exactly one tangent line can be drawn such that it is perpendicular to the radius at that point. [Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.]
3. Because there are infinitely many points $P_1, P_2, P_3, \dots, P_n$ on the circumference of the circle, there exist infinitely many corresponding tangent lines.
Conclusion:
As the number of points on the circumference is infinite, the number of tangents that can be drawn to a circle is also infinite.
Final Answer: A circle can have infinitely many tangents.
More Questions from Class 10 Mathematics Circles EXERCISE 10.1
- Q2(i): Fill in the blanks : (i) A tangent to a circle intersects it in __________ point (s).
- Q2(ii): Fill in the blanks : (ii) A line intersecting a circle in two points is called a __________.
- Q2(iii): Fill in the blanks : (iii) A circle can have __________ parallel tangents at the most.
- Q2(iv): Fill in the blanks : (iv) The common point of a tangent to a circle and the circle is called __________.
- 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 :
- 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.
CBSE Solutions for Class 10 Mathematics Circles
Chapters in CBSE - Class 10 Mathematics
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