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
Since PQ is tangent at P, OP ⊥ PQ (radius ⊥ tangent).
In right △OPQ:
$$PQ^2 = OQ^2 - OP^2 = 12^2 - 5^2 = 144 - 25 = 119$$$$PQ = \sqrt{119} \text{ cm}$$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
Construction:
- Draw a circle with centre O and a diameter AB.
- Draw the given line $l$ (parallel to AB direction).
- Draw a tangent at point A (perpendicular to OA, parallel to the original line if original line is perpendicular to OA) — this is a tangent parallel to the given line.
- Draw a secant cutting through the circle at two points, parallel to the first tangent — this is a secant parallel to the given line.
The two parallel lines: one tangent (touching the circle at exactly one point) and one secant (intersecting the circle at two points).
Note: For any diameter, there exist exactly two parallel tangents — one at each endpoint of the diameter. A secant parallel to these tangents would pass through the interior of the circle.