Class 10 Maths circles questions with solutions

Q 1 :

Two circles touch each other externally at C and AB is common tangent of circles, then $∠ A C B$ is

70°
60°
100°
90°

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(4)

Draw CM perpendicular to AB.

Now, AM = MC and MB = MC (tangents drawn from external point are equal).

⇒ AM = MC

⇒ $∠ M A C$ = $∠ M C A$ = 45°

(Since Δ AMC is right triangle)

∴ Also, MB = MC ⇒ $∠ M B C$ = $∠ M C B$ = 45° (Since Δ MBC is right angle triangle)

∴ $∠ A C B$ = $∠ M C A$ + $∠ M C B$ = 45° + 45° = 90° ⇒ $∠ A C B$ = 90°

Q 2 :

If the circumference of a circle and the perimeter of a square are equal, then

Area of the circle = Area of the square
Area of the circle > Area of the square
Area of the circle < Area of the square
Nothing definite can be said about the relation between the areas of the circle and square.

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(2)

Area of the circle > Area of the square

Q 3 :

The area of the circle that can be inscribed in a square of 6cm is

$36 π {cm}^{2}$
$18 π {cm}^{2}$
$12 π {cm}^{2}$
$9 π {cm}^{2}$

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4

(4)

ABCD is a square of side 6 cm. PQ is a diameter of the given circle such that $P Q = A B = 6 c m$

$∴$ $Radius (r) = \frac{6}{2} = 3 cm$

$Area of the circle = π r^{2} = π {(3)}^{2} = 9 π {cm}^{2} .$

Q 4 :

If two tangents inclined at an angle of $60^{o}$ are drawn to a circle of radius 3cm, then the length of each tangent is equal to

$3 \sqrt{3} / 2 cm$
$3 cm$
$6 cm$
$3 \sqrt{3} cm$

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(4)

Angle between two tangents = $60 °$ (given)

$∴$ Tangents are equally inclined to each other

$∴$ $∠ O P A = ∠ O P B = 30 °$

and $∠ O A P = 90 °$

(Tangent is perpendicular to the radius)

In $∆ P A O,$ $\tan 30 ° = \frac{O A}{A P} \Rightarrow \frac{1}{\sqrt{3}} = \frac{3}{A P} \Rightarrow A P = 3 \sqrt{3}$

Hence, the length of each tangent is $3 \sqrt{3}$ cm

Q 5 :

In the given figure, tangents PA and PB to the circle centred at O, from point P are perpendicular to each other. If PA = 5 cm, then length of AB is equal to

$5$ cm
$5 \sqrt{2}$ cm
$2 \sqrt{5}$ cm
$10$ cm

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(2)

PA = PB (Tangents from an external point P)

$∆ APB$ is a Right angle Triangle

$A B^{2} = A P^{2} + P B^{2}$ $[AP = PB]$

$A B^{2} = 2 A P^{2}$

$A B^{2} = 2 \times 5^{2}$

$\Rightarrow A B = 5 \sqrt{2} cm$

Q 6 :

In the given figure, PA and PB are two tangents drawn to the circle with centre O and radius 5 cm. If $∠ A P B = 60 °$ , then the length of PA is :

$5 / \sqrt{3} c m$
$5 \sqrt{3} c m$
$10 / \sqrt{3} c m$
$10 c m$

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4

(2) $5 \sqrt{3} c m$

Q 7 :

AB and CD are two chords of a circle intersecting at P. Choose the correct statement from the following:

$∆ A D P ~ ∆ C B A$
$∆ A D P ~ ∆ B P C$
$∆ A D P ~ ∆ B C P$
$∆ A D P ~ ∆ C B P$

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(4)

$∠ A P D = ∠ C P B$ [vertically opposite angle]

$∠ A D P = ∠ C B P$ [Angle subtends on the same segment]

$∆ A D P ~ ∆ C B P$ (by AA similarity)

Q 8 :

In the given figure, AT is tangent to a circle centred at O. If $∠ C A T = 40 °$ , then $∠ C B A$ is equal to

70°
50°
65°
40°

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(4)

Given, $∠ C A T = 40 °$

$∠ B A T = 90 °$

$\Rightarrow ∠ B A C + ∠ C A T = 90 °$

$\Rightarrow ∠ B A C = 50 °$

$\Rightarrow ∠ A C B = 90 ° [Angle in semi-circle]$

$In ∆ A B C, ∠ A + ∠ B + ∠ C = 180 °$

$\Rightarrow 50 ° + ∠ B + 90 ° = 180 °$

$\Rightarrow ∠ B = 180 ° - 140 ° = 40 °$

Q 9 :

Maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is:

4
3
2
1

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3

4

(3)

The maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is 2, as shown in

Q 10 :

In Fig, O is the centre of the circle. MN is the chord and the tangent ML at point M makes an angle of 70° with MN. The measure of ∠MON is:

120°
140°
70°
90°

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(2)

As ML is tangent at M and OM is radius of the circle, therefore ∠OML = 90°

⇒ ∠OMN + ∠NML = 90°
⇒ ∠OMN + 70° = 90° ⇒ ∠OMN = 20°

In ΔOMN, we have OM = ON (? radii of circle)
⇒ ∠OMN = ∠ONM = 20° (Angles opposite to equal sides are equal)

Also, ∠OMN + ∠MON + ∠ONM = 180°
⇒ 20° + ∠MON + 20° = 180°
⇒ ∠MON = 180° – 40° = 140°

Topic Question Set

Q 1 :

Two circles touch each other externally at C and AB is common tangent of circles, then $∠ A C B$ is

Q 2 :

If the circumference of a circle and the perimeter of a square are equal, then

Q 3 :

The area of the circle that can be inscribed in a square of 6cm is

Q 4 :

If two tangents inclined at an angle of $60^{o}$ are drawn to a circle of radius 3cm, then the length of each tangent is equal to

Q 5 :

In the given figure, tangents PA and PB to the circle centred at O, from point P are perpendicular to each other. If PA = 5 cm, then length of AB is equal to

Q 6 :

In the given figure, PA and PB are two tangents drawn to the circle with centre O and radius 5 cm. If $∠ A P B = 60 °$ , then the length of PA is :

Q 7 :

AB and CD are two chords of a circle intersecting at P. Choose the correct statement from the following:

Q 8 :

In the given figure, AT is tangent to a circle centred at O. If $∠ C A T = 40 °$ , then $∠ C B A$ is equal to

Q 9 :

Maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is:

Q 10 :

In Fig, O is the centre of the circle. MN is the chord and the tangent ML at point M makes an angle of 70° with MN. The measure of ∠MON is:

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Topic Question Set

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Q 2 :

If the circumference of a circle and the perimeter of a square are equal, then

Q 3 :

The area of the circle that can be inscribed in a square of 6cm is