CBSE Class 10 Circles Sure Shot Questions Set 02

Multiple Choice Questions

Question. A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is
(a) \( 5\sqrt{2} \)
(b) \( 10\sqrt{2} \)
(c) \( \frac{5}{\sqrt{2}} \)
(d) \( 10\sqrt{3} \)
Answer: (b)

Question. The length of the tangent drawn from a point 8 cm away from the centre of circle of radius 6 cm is
(a) \( \sqrt{7} \) cm
(b) \( 2\sqrt{7} \) cm
(c) 10 cm
(d) 5 cm
Answer: (b)

Question. In figure, PQ is tangent to the circle with centre at O, at the point B. If \( \angle AOB = 100^\circ \), then \( \angle ABP \) is equal to
(a) 50°
(b) 40°
(c) 60°
(d) 80°
Answer: (a)

Question. From a point Q, the length of the tangent to a circle is 12 cm and the distance of Q from the centre is 15 cm. The radius of the circle is
(a) 9 cm
(b) 12 cm
(c) 15 cm
(d) 24.5 cm
Answer: (a)

Question. In figure, if \( \angle AOB = 125^\circ \), then \( \angle COD \) is equal to
(a) 62.5°
(b) 45°
(c) 35°
(d) 55°
Answer: (d)

Question. Two concentric circles of radii 13 cm and 5 cm are given. The length of the chord of the larger circle which touches the smaller circle is
(a) 16 cm
(b) 4 cm
(c) 24 cm
(d) 10 cm
Answer: (c)

Question. Two concentric circles of radii \( a \) and \( b \) where \( a > b \), are given, the length of a chord of the larger circle which touches the other circle is
(a) \( \sqrt{a^2 - b^2} \)
(b) \( 2\sqrt{a^2 - b^2} \)
(c) \( \sqrt{a^2 + b^2} \)
(d) \( 2\sqrt{a^2 + b^2} \)
Answer: (b)

Question. 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
Answer: (d)

Question. A tangent to a circle is a line that touches the circle at exactly
(a) two points
(b) three points
(c) one point
(d) none of these
Answer: (c)

Question. If the angle between two radii of a circle is 140°, then the angle between the tangents at the ends of the radii is
(a) 90°
(b) 50°
(c) 70°
(d) 40°
Answer: (d)

Question. The radii of two concentric circles are 16 cm and 10 cm. AB is a diameter of the bigger circle. BD is tangent to the smaller circle touching it at D. Find the length of AD.
(a) \( 3\sqrt{130} \) cm
(b) \( 2\sqrt{139} \) cm
(c) \( 2\sqrt{130} \) cm
(d) \( 4\sqrt{139} \) cm
Answer: (b)

Question. If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to
(a) \( \frac{3}{2}\sqrt{3} \) cm
(b) 6 cm
(c) 3 cm
(d) \( 3\sqrt{3} \) cm
Answer: (d)

Question. How many tangents can a circle have from a point lying inside the circle ?
(a) 2
(b) infinitely many
(c) 1
(d) none of these
Answer: (d)

Question. Two parallel lines touch the circle at points A and B. If area of the circle is \( 16\pi \) \( \text{cm}^2 \), then AB is equal to
(a) 5 cm
(b) 8 cm
(c) 10 cm
(d) 16 cm
Answer: (b)

Question. If four sides of a quadrilateral ABCD are tangent to a circle, then
(a) AC + AD = BD + CD
(b) AB + CD = BC + AD
(c) AB + CD = AC + BC
(d) AC + AD = BC + DB
Answer: (b)

Question. In the diagram, PQ and QR are tangents to the circle with centre O, at P and R respectively. Find the value of \( x \).
(a) 25°
(b) 35°
(c) 45°
(d) 55°
Answer: (a)

Question. From an external point A, two tangents AB and AC are drawn to the circle with centre O. Then OA is the perpendicular bisector of
(a) BC
(b) AB
(c) AC
(d) none of these
Answer: (a)

Question. If the radii of two concentric circles are 6 cm and 10 cm, the length of chord of the larger circle which is tangent to other is
(a) 14 cm
(b) 16 cm
(c) 18 cm
(d) 12 cm
Answer: (b)

Question. Assertion (A): When two tangents are drawn to a circle from an external point, they subtend equal angles at the centre.
Reason (R): A parallelogram circumscribing a circle is a rhombus.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b)

Question. Assertion (A): If in a cyclic quadrilateral, one angle is 40°, then the opposite angle is 140°.
Reason (R): Sum of opposite angles in a cyclic quadrilateral is equal to 360°.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (c)

Question. Assertion (A): PA and PB are two tangents to a circle with centre O such that \( \angle AOB = 110^\circ \), then \( \angle APB = 90^\circ \).
Reason (R): The length of two tangents drawn from an external point are equal.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (d)

Question. Two concentric circles of radii \( a \) and \( b \) (\( a > b \)) are given. Find the length of the chord of the larger circle which touches the smaller circle.
Answer: \( 2\sqrt{a^2 - b^2} \)

Question. From an external point P, tangents PA and PB are drawn to a circle with centre O. If \( \angle PAB = 50^\circ \), then find \( \angle AOB \).
Answer: In \( \triangle PAB \), PA = PB (tangents from external point). So, \( \angle PBA = \angle PAB = 50^\circ \).
\( \angle APB = 180^\circ - (50^\circ + 50^\circ) = 80^\circ \).
In quadrilateral AOBP, \( \angle AOB = 180^\circ - 80^\circ = 100^\circ \).

Question. If the angle between two tangents drawn from an external point ‘P’ to a circle of radius ‘r’ and centre O is 60°, then find the length of OP.
Answer: Angle between tangents \( \angle APB = 60^\circ \). OP bisects this angle, so \( \angle OPA = 30^\circ \).
In right \( \triangle OAP \), \( \sin 30^\circ = \frac{OA}{OP} \Rightarrow \frac{1}{2} = \frac{r}{OP} \Rightarrow OP = 2r \).

Question. If the radii of two concentric circles are 4 cm and 5 cm, then find the length of each chord of one circle which is tangent to the other circle.
Answer: \( \text{Length} = 2\sqrt{5^2 - 4^2} = 2\sqrt{9} = 6 \text{ cm} \).

Question. PQ is a tangent to a circle with centre O at point P. If \( \triangle OPQ \) is an isosceles triangle, then find \( \angle OQP \).
Answer: Since PQ is tangent at P, \( \angle OPQ = 90^\circ \).
In isosceles \( \triangle OPQ \), \( \angle POQ = \angle OQP \).
Sum of angles \( = 180^\circ \Rightarrow 2 \angle OQP + 90^\circ = 180^\circ \Rightarrow \angle OQP = 45^\circ \).

Question. Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.
Answer: Let AB and CD be two parallel tangents to a circle with centre O, touching it at P and Q respectively. Draw a line OE parallel to AB. Since radius is perpendicular to tangent, \( \angle OPA = 90^\circ \). Since AB || OE, \( \angle POE = 180^\circ - 90^\circ = 90^\circ \). Similarly, \( \angle QOE = 90^\circ \). Now \( \angle POQ = \angle POE + \angle QOE = 90^\circ + 90^\circ = 180^\circ \). Thus POQ is a straight line passing through O.

Case Based MCQs

Case I : Read the following passage and answer the questions.

Smita always finds it confusing with the concepts of tangent and secant of a circle. But this time she has determined herself to get concepts easier. So, she started listing down the differences between tangent and secant of a circle along with their relation. Here, some points in question form are listed by Smita in her notes.

Question. A line that intersects a circle exactly at two points is called
(a) Secant
(b) Tangent
(c) Chord
(d) Both (a) and (b)
Answer: (a)

Question. Number of tangents that can be drawn on a circle is
(a) 1
(b) 0
(c) 2
(d) Infinite
Answer: (d)

Question. Number of tangents that can be drawn to a circle from a point not on it, is
(a) 1
(b) 2
(c) 0
(d) Infinite
Answer: (b)

Question. Number of secants that can be drawn to a circle from a point on it is
(a) infinite
(b) 1
(c) 2
(d) 0
Answer: (a)

Question. A line that touches a circle at only one point is called
(a) Secant
(b) Chord
(c) Tangent
(d) Diameter
Answer: (c)

Case II : Read the following passage and answer the questions.

Theorem on Circles

If a tangent is drawn to a circle from an external point, then the radius at the point of contact is perpendicular to the tangent.

Question. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
(a) 8 cm
(b) 4 cm
(c) 10 cm
(d) 6 cm
Answer: (a)

Question. Two concentric circles are such that the difference between their radii is 4 cm and the length of the chord of the larger circle which touches the smaller circle is 24 cm. Then the radius of the smaller circle is
(a) 16 cm
(b) 20 cm
(c) 18 cm
(d) None of these
Answer: (d)

Assertion & Reasoning Based MCQs

Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices :
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion : If TP and TQ are the two tangents to a circle with centre O so that \( \angle POQ = 123^\circ \), then \( \angle PTQ = 57^\circ \).
Reason : The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Answer: (a)

Question. Assertion : At a point P of a circle with centre O and radius 12 cm, a tangent PQ of length 16 cm is drawn. Then, OQ = 20 cm.
Reason : The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Answer: (a)

Question. Assertion : The secant of circle is perpendicular to the radius of the circle.
Reason : A line that intersects the given circle in two points is called a secant.
Answer: (b)

Question. Assertion : The length of tangents drawn from an external point to a circle are not always equal in length.
Reason : The tangent is always perpendicular to the radius through the point of contact.
Answer: (d)

Very Short Answer Type Questions 

Question. If the angle between two tangents drawn from an external point P to a circle of radius \( a \) and centre O, is 60°, then find the length of OP.
Answer: Let AP and BP be tangents.
In \( \Delta OAP \), \( \angle OAP = 90^\circ \) and \( \angle APO = 30^\circ \)
\( \sin 30^\circ = \frac{OA}{OP} \)
\( \frac{1}{2} = \frac{a}{OP} \)
OP = \( 2a \)