CBSE Class 10 Mathematics Circles MCQs Set 06

Question. Two chords \( AB \) and \( CD \) of a circle intersect at \( E \) such that \( AE = 2.4 \text{ cm} \), \( BE = 3.2 \text{ cm} \) and \( CE = 1.6 \text{ cm} \). The length of \( DE \) is
(a) 1.6 cm
(b) 3.2 cm
(c) 4.8 cm
(d) 6.4 cm
Answer: C

Question. If a regular hexagon is inscribed in a circle of radius \( r \), then its perimeter is
(a) \( 3r \)
(b) \( 6r \)
(c) \( 9r \)
(d) \( 12r \)
Answer: B
Side of the regular hexagon inscribed in a circle of radius \( r \) is also \( r \), the perimeter is \( 6r \).

Question. Two circles of radii 20 cm and 37 cm intersect in \( A \) and \( B \). If \( O_1 \) and \( O_2 \) are their centres and \( AB = 24 \text{ cm} \), then the distance \( O_1 O_2 \) is equal to
(a) 44 cm
(b) 51 cm
(c) 40.5 cm
(d) 45 cm
Answer: B

Question. In two concentric circles, if chords are drawn in the outer circle which touch the inner circle, then
(a) all chords are of different lengths.
(b) all chords are of same length.
(c) only parallel chords are of same length.
(d) only perpendicular chords are of same length.
Answer: B

Question. Number of tangents to a circle which are parallel to a secant, is
(a) 3
(b) 2
(c) 1
(d) infinite
Answer: B
Only two tangents are parallel to a secant.

Question. \( AB \) and \( CD \) are two common tangents to circles which touch each other at a point \( C \). If \( D \) lies on \( AB \) such that \( CD = 4 \text{ cm} \) then \( AB \) is
(a) 12 cm
(b) 8 cm
(c) 4 cm
(d) 6 cm
Answer: B

Question. An equation of the circle with centre at (0, 0) and radius \( r \) is
(a) \( x^2 + y^2 = r^2 \)
(b) \( x^2 - y^2 = r^2 \)
(c) \( x - y = r \)
(d) \( x^2 + r^2 = y^2 \)
Answer: A
Here, \( h = k = 0 \). Therefore, the equation of the circle is \( x^2 + y^2 = r^2 \).

FILL IN THE BLANK

Question. .......... is the Latin word from which the word tangent has been derived.
Answer: Tangere

Question. The common point of a tangent to a circle and the circle is called ..........
Answer: point of contact

Question. There is no tangent to a circle passing through a point lying .......... the circle.
Answer: inside

Question. The tangent to a circle is .......... to the radius through the point of contact.
Answer: perpendicular

Question. There are exactly two tangents to a circle passing through a point lying .......... the circle.
Answer: outside

Question. Length of two tangents drawn from an external point are ..........
Answer: equal

TRUE/FALSE

Question. The perpendicular at the point of contact to the tangent to a circle does not pass through the centre.
Answer: False

Question. The length of tangent from an external point on a circle is always greater than the radius of the circle.
Answer: True

Question. A circle can have at the most two parallel tangents.
Answer: True

Question. If a chord \( AB \) subtends an angle of \( 60^{\circ} \) at the centre of a circle, then angle between the tangents at \( A \) and \( B \) is also \( 60^{\circ} \).
Answer: False

Question. If \( P \) is a point on a circle with centre \( C \), then the line drawn through \( P \) and perpendicular to \( CP \) is the tangent to the circle at the point \( P \).
Answer: True

Question. If a number of circles touch a given line segment \( PQ \) at a point \( A \), then theri centres lie on the perpendicular

Question. bisector of \( PQ \).
Answer: False

Question. The centre of the circle lies on the bisector of the angle between the two tangents.
Answer: True

Question. Two equal chords of a circle are always parallel.
Answer: False

Question. \( AB \) is a diameter of a circle and \( AC \) is its chord such that \( \angle BAC = 30^\circ \). If the tangent at \( C \) intersects \( AB \) extended at \( D \), then \( BC = BD \).
Answer: True

MATCHING QUESTIONS

DIRECTION : Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in Column-I have to be matched with statements (p, q, r, s) in Column-II.

Question. Match the following terms related to circles:
Column-I
(A) A line segment which join any two points on a circle.
(B) A line which intersect the circle in two points.
(C) A line that intersects the circle at only one point.
Column-II
(p) Secant
(q) Tangent
(r) Chord
Answer: (A) - r, (B) - p, (C) - q

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(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.

Question. Assertion : The two tangents are drawn to a circle from an external point, then they subtend equal angles at the centre.
Reason : 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 : If in a cyclic quadrilateral, one angle is \( 40^\circ \), then the opposite angle is \( 140^\circ \).
Reason : Sum of opposite angles in a cyclic quadrilateral is equal to \( 360^\circ \).
(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 : \( PA \) and \( PB \) are two tangents to a circle with centre \( O \). Such that \( \angle AOB = 110^\circ \), then \( \angle APB = 90^\circ \).
Reason : 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