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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
Top Tutors who teach Circles
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Electronic City Phase I, Bangalore
20 yrs of
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842 hrs of classes completed
400
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Class 10 is a pivotal, fast-paced school year filled with a mix of academic preparation, growing responsibilities, and memorable moments with friends.Daily Routine and StudiesBusy schedules: Days involve attending subject-specific classes, taking detailed notes, and preparing for board exams.Favorite subjects: Time is often spent engaging deeply with core subjects like mathematics, science, and social studies.Extra help: Teachers provide extra guidance and revision sessions to help students handle exam stress.Friendships and Social LifeStronger bonds: Friendships deepen through shared lunch breaks, group studies, and mutual support during exams.Lasting memories: Small moments in the hallway or sharing stories in the classroom create a sense of a second family.
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Nanded, Pune
12 yrs of
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1000
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Dedicated and passionate educator with over 10 years of experience teaching Mathematics and Science to students of Classes 8th to 12th across multiple boards — CBSE, ICSE, IGCSE, IB MYP, and IBDP. Founder of Ishaan Tuitions, Pune, specializing in personalized one-to-one coaching and small group sessions designed to strengthen conceptual understanding and academic performance. Expertise also includes preparing students for competitive examinations such as SAT and IIT-JEE Mains, with a strong record of improving analytical thinking and problem-solving abilities.
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Saheed Nagar, Bhubaneswar
11 yrs of
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36 students
640 hrs of classes completed
499
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Mohali SAS Nagar, Chandigarh
15 yrs of
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18 students
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West Marredpally, Hyderabad
20 yrs of
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I have more than 15 years teaching experience in teaching CBSE classes 8-10 Physics/Chemistry. I can also teach math upto 9th . All my students secured 10/10 in previous years exams. And in school more than 90% students got more than 90 in Science Exam and all have passed in Science i. Physics and Chemistry.
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