Get complete, step-by-step solutions for NCERT Class 10 Maths Chapter 10 Exercise 10.1 on Circles. Solutions cover the fundamental concepts of tangents and secants, including defining a tangent and a secant, identifying the point of contact, and determining the maximum number of parallel tangents a circle can have (Q.1, Q.2). Practice applying the property that the radius is perpendicular to the tangent at the point of contact to calculate the length of a tangent using the Pythagorean theorem (Q.3). Includes instructions for drawing a circle with parallel tangent and secant lines (Q.4). Essential for building a strong foundation in circle geometry.
1. How many tangents can a circle have?
A circle can have infinitely many tangents.
Justification: A tangent touches the circle at exactly one point, and there are infinitely many points on the circumference of a circle.
2. 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.
(iv) The common point of a tangent to a circle and the circle is called the point of contact.
3. Length PQ
Let $P$ be the point of contact, $O$ be the center, and $PQ$ be the tangent.
The radius $OP = 5 \text{ cm}$. The distance from the center to $Q$ is $OQ = 12 \text{ cm}$.
Since the radius is perpendicular to the tangent at the point of contact, $\triangle OPQ$ is a right-angled triangle with the right angle at $P$ ($\angle OPQ = 90^\circ$). $OQ$ is the hypotenuse.
Using the Pythagorean theorem:
$$OP^2 + PQ^2 = OQ^2$$
$$5^2 + PQ^2 = 12^2$$
$$25 + PQ^2 = 144$$
$$PQ^2 = 144 – 25$$
$$PQ^2 = 119$$
$$PQ = \sqrt{119} \text{ cm}$$
The correct option is (D) $\sqrt{119} \text{ cm}$.
(Note: The textbook options often write the number under the root without the root sign as a typo, but $\sqrt{119}$ is the correct answer based on the calculation.)
4. Draw a circle and two lines parallel to a given line
- Step 1: Draw a circle with center $O$ and any radius $r$.
- Step 2: Draw a given line $l$.
- Step 3: Draw a line $m$ parallel to $l$ that passes through two points of the circle (a secant).
- Step 4: Draw a line $n$ parallel to $l$ that touches the circle at exactly one point, say $A$ (a tangent).
