Circles : Exercise 10.1 (Mathematics NCERT Class 10th)

Q.1      How many tangents can a circle have ?
Sol.         A circle can have an infinite number of tangents.

Q.2      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 ______.
Sol.         (i) exactly one
(ii) secant
(iii) two
(iv) point of contact

Q.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 of PQ is :
(A) 12 cm                  (B) 13 cm                   (C) 8.5 cm                    (D) $\sqrt {119} cm$
Sol.         (D). Because,
$PQ = \sqrt {O{Q^2} - O{P^2}} = \sqrt {{{12}^2} - {5^2}} = \sqrt {144 - 25}$
$= \sqrt {119} \,cm$

Q.4      Draw a circle and two lines parallel to a given line such that one is tangent and the other, a secant to the circle.
Sol.        The required figure is as under :
Here $\ell$ is the given line and a circle with centre O is drawn.
Line PT is drawn || to line $\ell$.
PT is the tangent to the circle.
AB is drawn || to line $\ell$ and is the secant.

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