Exercise-10.1, Class 10th, Maths, Chapter 10, NCERT

Q1. How many tangents can a circle have?

Solution.
A circle has infinitely many tangents.
Reason: For every point on the circle there is exactly one tangent at that point, and since a circle has infinitely many points, it admits infinitely many tangents.

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Q2. 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 ___.

Solution.
(i) one point.
(ii) a secant.
(iii) two. (At most two distinct lines tangent to a circle can be parallel.)
(iv) the point of contact (or simply the point of contact / point of tangency).

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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 :

(A) 12 cm (B) 13 cm (C) 8.5 cm (D) $\sqrt{119}$ cm.

Solution.
Draw radius $OP$. Radius $OP$ is perpendicular to the tangent at $P$ (Theorem 10.1), so triangle $OPQ$ is right-angled at $P$. Thus by Pythagoras,

$$P{Q}^2=O{Q}^2-O{P}^2={12}^2-5^2=144-25=119.$$

So $PQ=\sqrt{119}$ cm. Choice (D) is correct.

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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.

Solution / Construction (steps):

1. Draw any circle with centre $O$.

2. Draw a straight reference line $l$ somewhere near the circle (this is the “given line” whose parallels we will construct).

3. Through a point outside the circle, draw a line $m$ parallel to $l$ so that $m$ just touches the circle at exactly one point — adjust $m$ until it becomes tangent. (Equivalently: draw the line through the intended tangency point that is perpendicular to the radius at that point; make it parallel to $l$ by sliding if needed.) This line $m$ is the tangent(it meets the circle in exactly one point).

4. Draw another line $n$ parallel to $l$ but closer to the circle so that it cuts the circle in two points — this line $n$ is a secant (it meets the circle in two points).

Remark / short drawing hint: choose a point $T$ on the circle, draw the radius $OT$, then draw the line through $T$perpendicular to $OT$ — that line is a tangent. Now draw a line parallel to that tangent but shifted inward so it meets the circle in two points — that is a parallel secant.