NCERT Solutions for Class 10 Mathematics Chapter 10 Circles

Q1

Define a tangent and a secant of a circle. How many tangents can a circle have?

A tangent to a circle is a line that intersects the circle at exactly one point (called the point of contact). A secant is a line that intersects the circle at two distinct points. A circle has infinitely many tangents — exactly one tangent can be drawn at each point on the circumference.

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Q2

State and prove Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Proof: Let

O

be the centre,

XY

be the tangent at point

P

, and

OP

be the radius. Take any point

Q

on

XY

other than

P

. Join

OQ

. Since

Q

lies outside the circle,

OQ > OP

(radius). As this holds for every point on

XY

except

P

,

OP

is the shortest distance from

O

to the line

XY

. The shortest distance from a point to a line is the perpendicular distance. Hence,

OP \perp XY

.

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Q3

State and prove Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

Statement: The lengths of tangents drawn from an external point to a circle are equal. Proof: Let

P

be an external point and

PQ

,

PR

be tangents from

P

to a circle with centre

O

. Join

OP

,

OQ

,

OR

. By Theorem 10.1,

\angle OQP = \angle ORP = 90^\circ

. In right triangles

OQP

and

ORP

,

OQ = OR

(radii) and

OP = OP

(common). By RHS congruence,

\triangle OQP \cong \triangle ORP

. By CPCT,

PQ = PR

. Remarks:

\angle OPQ = \angle OPR

, so

OP

is the angle bisector of

\angle QPR

. Also, using Pythagoras:

PQ^2 = OP^2 - OQ^2 = OP^2 - OR^2 = PR^2

, giving

PQ = PR

.

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Q4

How many tangents can be drawn from a point to a circle when the point is (i) inside the circle, (ii) on the circle, and (iii) outside the circle? Explain with reasoning.

(i) Point inside the circle: No tangent can be drawn. Every line through an interior point is a secant (intersects the circle at two points). (ii) Point on the circle: Exactly one tangent can be drawn. By Theorem 10.1, only one line through a point on the circle can be perpendicular to the radius. (iii) Point outside the circle: Exactly two tangents can be drawn. These are the two lines from the external point that touch the circle.

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Q5

Prove that in two concentric circles, the chord of the larger circle which touches the smaller circle is bisected at the point of contact.

Let two concentric circles

C_1

(larger) and

C_2

(smaller) have common centre

O

. Let chord

AB

of

C_1

touch

C_2

at point

P

. Join

OP

. Since

AB

is a tangent to

C_2

at

P

, by Theorem 10.1,

OP \perp AB

. Now

OP

is the perpendicular from the centre

O

to the chord

AB

of

C_1

. The perpendicular from the centre of a circle to a chord bisects the chord. Therefore,

AP = BP

.

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Q6

Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$ . Prove that $\angle PTQ = 2\angle OPQ$ .

Let

\angle PTQ = \theta

. By Theorem 10.2,

TP = TQ

, so

\triangle TPQ

is isosceles. Thus

\angle TPQ = \angle TQP = \frac{1}{2}(180^\circ - \theta) = 90^\circ - \frac{\theta}{2}

. By Theorem 10.1,

\angle OPT = 90^\circ

. Then

\angle OPQ = \angle OPT - \angle TPQ = 90^\circ - (90^\circ - \frac{\theta}{2}) = \frac{\theta}{2}

. Therefore,

\angle PTQ = \theta = 2\angle OPQ

.

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Q7

$PQ$ is a chord of length $8\text{ cm}$ of a circle of radius $5\text{ cm}$ . The tangents at $P$ and $Q$ intersect at a point $T$ . Find the length $TP$ .

Join

OT

. Let

OT

meet

PQ

at

R

. Since

\triangle TPQ

is isosceles and

TO

is the angle bisector of

\angle PTQ

,

OT \perp PQ

and

PR = RQ = 4\text{ cm}

. In right

\triangle OPR

,

OR = \sqrt{OP^2 - PR^2} = \sqrt{5^2 - 4^2} = 3\text{ cm}

. By AA similarity, right

\triangle TRP \sim \triangle PRO

, giving

\frac{TP}{PO} = \frac{RP}{RO}

, i.e.,

\frac{TP}{5} = \frac{4}{3}

, so

TP = \frac{20}{3}\text{ cm}

.

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Circles

All questions

Define a tangent and a secant of a circle. How many tangents can a circle have?

State and prove Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

State and prove Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

How many tangents can be drawn from a point to a circle when the point is (i) inside the circle, (ii) on the circle, and (iii) outside the circle? Explain with reasoning.

Prove that in two concentric circles, the chord of the larger circle which touches the smaller circle is bisected at the point of contact.

Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$ . Prove that $\angle PTQ = 2\angle OPQ$ .

$PQ$ is a chord of length $8\text{ cm}$ of a circle of radius $5\text{ cm}$ . The tangents at $P$ and $Q$ intersect at a point $T$ . Find the length $TP$ .

Other Chapters

Circles

All questions

Define a tangent and a secant of a circle. How many tangents can a circle have?

State and prove Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

State and prove Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

How many tangents can be drawn from a point to a circle when the point is (i) inside the circle, (ii) on the circle, and (iii) outside the circle? Explain with reasoning.

Prove that in two concentric circles, the chord of the larger circle which touches the smaller circle is bisected at the point of contact.

Two tangents TPTPTP and TQTQTQ are drawn to a circle with centre OOO from an external point TTT. Prove that ∠PTQ=2∠OPQ\angle PTQ = 2\angle OPQ∠PTQ=2∠OPQ.

PQPQPQ is a chord of length 8 cm8\text{ cm}8 cm of a circle of radius 5 cm5\text{ cm}5 cm. The tangents at PPP and QQQ intersect at a point TTT. Find the length TPTPTP.

Other Chapters

Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$ . Prove that $\angle PTQ = 2\angle OPQ$ .

$PQ$ is a chord of length $8\text{ cm}$ of a circle of radius $5\text{ cm}$ . The tangents at $P$ and $Q$ intersect at a point $T$ . Find the length $TP$ .