NCERT Exemplar Class 10 Maths Solutions Chapter 9 Circles

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NCERT Exemplar Problems Class 10 Maths Solutions Chapter 9 Circles

Exercise 9.1 Multiple Choice Questions (MCQs)

Question 1:
If radii of two concentric circles are 4 cm and 5 cm, then length of each chord of one circle which is tangent to the other circle, is
(a) 3 cm           (b) 6 cm                    (c) 9 cm                     (d) 1 cm
Solution:
(b) Let 0 be the centre of two concentric circles C₁ and C₂, whose radii are r₁ = 4 cm and r₂ = 5 cm. Now, we draw a chord AC of circle C₂, which touches the circle C₁ at B.
Also, join OB, which is perpendicular to AC. [Tangent at any point of circle is perpendicular to radius throughly the point of contact]

Question 2:
In figure, if ∠AOB = 125°, then ∠COD is equal to

(a) 62.5°               (b) 45°                    (c) 35°                     (d) 55°
Solution:
(d) We know that, the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Question 3:
In figure, AB is a chord of the circle and AOC is its diameter such that ∠ACB = 50°. If AT is the tangent to the circle at the point A, then ∠BAT is equal to

(a) 45°               (b) 60°                    (c) 50°                     (d) 55°
Solution:
(c) In figure, AOC is a diameter of the circle. We know that, diameter subtends an angle 90° at the circle.

Question 4:
From a point P which is at a distance of 13 cm from the centre 0 of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle is drawn. Then, the area of the quadrilateral PQOR is
(a) 60 cm²                  (b) 65 cm²                (c) 30 cm²                 (d) 32.5 cm²
Solution:
(a) Firstly, draw a circle of radius 5 cm having centre O. P is a point at a distance of 13 cm from O. A pair of tangents PQ and PR are drawn.

Thus, quadrilateral POOR is formed.

Question 5:
At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A, is
(a) 4 cm                                                    (b) 5 cm
(c) 6 cm                                                    (d) 8 cm
Solution:
(d) First, draw a circle of radius 5 cm having centre 0. A tangent XY is drawn at point A.

Question 6:
In figure, AT is a tangent to the circle with centre 0 such that OT = 4 cm and ∠OTA = 30°. Then, AT is equal to

(a) 4 cm               (b) 2 cm                   (c) 2√3 cm                     (d) 4√3 cm
Solution:
(c) Join OA
We know that, the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Question 7:
In figure, if 0 is the centre of a circle, PQ is a chord and the tangent PR at P , ; makes an angle of 50° with PQ, then ∠POQ is equal to

(a) 100°               (b) 80°                    (c) 90°                     (d) 75°
Solution:
(a) Given, ∠QPR = 50°
We know that, the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Question 8:
In figure, if PA and PB are tangents to the circle with centre 0 such that ∠APB = 50°, then ∠OAB is equal to

(a) 25°               (b) 30°                    (c) 40°                     (d) 50°
Solution:
(a) Given, PA and PB are tangent lines.

Question 9:
If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then the length of each tangent is
(a) \(\frac { 3 }{ 2 } \)√3 cm                   (b) 6 cm                (c) 3 cm                      (d) 3 √3 cm
Solution:
(d) Let P be an external point and a pair of tangents is drawn from point P and angle between these two tangents is 60°.

Tangent at any point of a circle is perpendicular to the radius through the point of contact.

Hence, the length of each tangent is 3√3 cm.

Question 10:
In figure, if PQR is the tangent to a circle at Q whose centre is 0, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to

(a) 20°               (b) 40°                    (c) 35°                     (d) 45°
Solution:

Exercise 9.2 Very Short Answer Type Questions

Question 1:
If a chord AB subtends an angle of 60° at the centre of a circle, then angle between the tangents at A and B is also 60°.
Solution:
False
Since a chord AB subtends an angle of 60° at the centre of a circle.

Question 2:
The length of tangent from an external point P on a circle is always greater than the radius of the circle.
Solution:
False
Because the length of tangent from an external point P on a circle may or may not be greater than the radius of the circle.

Question 3:
The length of tangent from an external point P on a circle with centre 0 is always less than OP.
Solution:
True

Question 4:
The angle between two tangents to a circle may be 0°.
Solution:
True
‘This may be possible only when both tangent lines coincide or are parallel to each other.

Question 5:
If angle between two tangents drawn from a point P  to a circle of radius a and centre 0 is 90°, then OP = a √2.
Solution:
True

Question 6:
If angle between two tangents drawn from a point P to a circle of radius a and centre 0 is 60°, then OP = a√3.
Solution:
True

Question 7:
The tangent to the circumcircle of an isosceles ΔABC at A, in which AB = AC, is parallel to BC.
Solution:
True
Let EAF be tangent to the circumcircle of ΔABC.

Question 8:
If a number of circles touch a given line segment PQ at a point A, then their centres lie on the perpendicular bisector of PQ.
Solution:
False
Given that PQ is any line segment and S_1, S₂, S₃, S₄,… circles are touch a line segment PQ at a point A. Let the centres of the circlesS_1,S₂, S₃, S₄,… be C₁ C₂, C₃, C₄,… respectively.

To prove centres of these circles lie on the perpendicular bisector PQ
Now, joining each centre of the circles to the point A on the line segment PQ by a line segment i.e., C₁A, C₂A, C₃A, C₄A… so on.
We know that, if we draw a line from the centre of a circle to its tangent line, then the line is always perpendicular to the tangent line. But it not bisect the line segment PQ.

Since, each circle is passing through a point A. Therefore, all the line segments
C₁A, C₂A, C₃A, C₄A…. so on are coincident.
So, centre of each circle lies on the perpendicular line of PQ but they do not lie on the perpendicular bisector of PQ.
Hence, a number of circles touch a given line segment PQ at a point A, then their centres lie

Question 9:
If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of PQ.
Solution:

Question 10:
AB is a diameter of a circle and AC is its chord such that ∠BAC – 30°. If the tangent at C intersects AB extended at D, then BC = BD.
Solution:
True
To Prove, BC = BD

Exercise 9.3 Short Answer Type Questions

Question 1:
Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.
Solution:
Let C₁ and C₂ be the two circles having same centre O. AC is a chord which touches the C₁ at point D.

Question 2:
Two tangents PQ and PR are drawn from an external point to a circle with centre 0. Prove that QORP is a cyclic quadrilateral.
Solution:
Given Two tangents PQ and PR are drawn from an external point to a circle with centre 0.

Question 3:
Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
Solution:
Given Two tangents PQ and PR are drawn from an external point P to a circle with centre 0.

To prove Centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
In ∠RPQ.
Construction Join OR, and OQ.
In ΔPOP and ΔPOO

Question 4:
If from an external point B of a circle with centre 0, two tangents BC and BD are drawn such that ∠DBC = 120°, prove that BC + BD = B0 i.e., BO = 2 BC.
Solution:
Two tangents BD and BC are drawn from an external point B.

Question 5:
In figure, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD

Solution:
Given AS and CD are common tangent to two circles of unequal radius
To prove                                        AB = CD

Question 6:
In figure, AB and CD are common tangents to two circles of equal radii. Prove that AB = CD.

Solution:
Given AB and CD are tangents to two circles of equal radii.
To prove                                               AB = CD

Question 7:
In figure, common tangents AB and CD to two circles intersect at E. Prove that AB = CD.

Solution:
Given Common tangents AB and CD to two circles intersecting at E.
To prove                                           AB = CD

Question 8:
A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.
Solution:
Given Chord PQ is parallel to tangent at R.
To prove R bisects the arc PRQ

Question 9:
Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.
Solution:
To prove ∠1 = ∠2, let PQ be a chord of the circle. Tangents are drawn at the points R and Q.

Let P be another point on the circle, then, join PQ and PR.
Since, at point Q, there is a tangent.

Question 10:
Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A.
Solution:
Given, AB is a diameter of the circle.
A tangent is drawn from point A. Draw a chord CD parallel to the tangent MAN.

Thus, OE bisects CD, [perpendicular from centre of circle to chord bisects the chord] Similarly, the diameter AB bisects all. Chords which are parallel to the tangent at the point A.

Exercise 9.4 Long Answer Type Questions

Question 1:
If a hexagon ABCDEF circumscribe a circle, prove that
AB + CD + EF =BC + DE + FA
Solution:
Given A hexagon ABCDEF circumscribe a circle.

Question 2:
Let s denotes the semi-perimeter of a Δ ABC in which BC = a, CA = b and AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively. Prove that BD = s – b.
Solution:
A circle is inscribed in the A ABC, which touches the BC, CA and AB.

Question 3:
From an external point P, two tangents, PA and PB are drawn to a circle with centre 0. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the trianlge PCD.
Solution:
Two tangents PA and PB are drawn to a circle with centre 0 from an external point P

Question 4:
If AB is a chord of a circle with centre 0, AOC is a diameter and AT is the tangent at A as shown in figure. Prove that ∠BAT = ∠ACB.

Solution:
Since, AC is a diameter line, so angle in semi-circle makes an angle 90°.

Question 5:
Two circles with centres 0 and 0′ of radii 3 cm and 4 cm, respectively intersect at two points P and Q, such that OP and 0’P are tangents to the two circles. Find the length of the common chord PQ.
Solution:
Here, two circles are of radii OP = 3 cm and PO’ = 4 cm
These two circles intersect at P and Q.

Question 6:
In a right angle ΔABC is which ∠B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at PQ bisects BC.
Solution:
Let O be the centre of the given circle. Suppose, the tangent at P meets BC at 0. Join BP.

Question 7:
In figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.
Solution:
PQ and PR are two tangents drawn from an external point P.

Question 8:
AB is a diameter and AC is a chord of a circle with centre 0 such that ∠BAC = 30°. The tangent at C intersects extended AB at a point D. Prove that BC = BD.
Solution:
A circle is drawn with centre O and AB is a diameter.
AC is a chord such that ∠BAC = 30°.
Given AB is a diameter and AC is a chord of circle with certre O, ∠BAC = 30°.

Question 9:
Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.
Solution:
Let mid-point of an arc AMB be M and TMT’ be the tangent to the circle.
Join AB, AM and MB.

But ∠AMT and ∠MAB are alternate angles, which is possible only when
AB\(\parallel \)TMT’
Hence, the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc
Hence proved.

Question 10:
In a figure the common tangents, AB and CD to two circles with centres 0 and O’ intersect at E. Prove that the points 0, E and O’ are collinear.

Solution:

Question 11:
In figure, 0 is the centre of a circle of radius 5 cm, T is a point such that OT = 13 and 0T intersects the circle at E, if AB is the tangent to the circle at E, find the length of AB.

Solution:
Given, OT = 13 cm and OP = 5 cm
Since, if we drawn a line from the centre to the tangent of the circle. It is always  perpendicular to the tangent i.e., OP⊥PT.

Question 12:
The tangent at a point C of a circle and a diameter AB when extended intersect at P. If ∠PCA = 110°, find ∠CBA.
Solution:
Here, AB is a diameter of the circle from point C and a tangent is drawn which meets at a point P.

Question 13:
If an isosceles ΔABC in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, find the area of the triangle.
Solution:
In a circle, ΔABC is inscribed.
Join OB, OC and OA.

Question 14:
A is a point at a distance 13 cm from the centre 0 of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ΔABC.
Solution:
Given Two tangents are drawn from an external point A to the circle with centre 0,

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