CBSE Class 10 Mathematics Circles MCQs Set E

Question. If the equation of a circle is \( (4a - 3)x^2 + ay^2 + 6x - 2y + 2 = 0 \), then its centre is
(a) \( (3, -1) \)
(b) \( (3, 1) \)
(c) \( (-3, 1) \)
(d) None of the options
Answer: C
Since the given equation represents a circle, therefore,
\( 4a - 3 = a \) i.e., \( a = 1 \)
(coefficients of \( x^2 \) and \( y^2 \) must be equal)
\( x^2 + y^2 + 6x - 2y + 2 = 0 \)
The coordinates of centre are \( (-3, 1) \).

Question. The common tangents to the circles \( x^2 + y^2 + 2x = 0 \) and \( x^2 + y^2 - 6x = 0 \) form a triangle which is
(a) equilateral
(b) isosceles
(c) right angled
(d) None of the options
​​
Answer: B

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: D

Question. The equation of the circle which passes through the point (4, 5) and has its centre at (2, 2) is
(a) \( (x - 2)^2 + (y - 2)^2 = 13 \)
(b) \( (x - 2)^2 + (y - 2)^2 = 13 \)
(c) \( x^2 + y^2 = 13 \)
(d) \( (x - 4)^2 + (y - 5)^2 = 13 \)
Answer: B
As the circle is passing through the point (4, 5) and its centre is (2, 2) so its radius is
\( \sqrt{(4-2)^2 + (5-2)^2} = \sqrt{13} \)
Therefore, the required equation is
\( (x - 2)^2 + (y - 2)^2 = 13 \)

Question. Two concentric circles are of radii 10 cm and 8 cm, then the length of the chord of the larger circle which touches the smaller circle is:
(a) 6 cm
(b) 12 cm
(c) 18 cm
(d) 9 cm
Answer: B

Question. A tangent \( PQ \) at a point \( P \) of a circle of radius \( 6 \text{ cm} \) meets a line through the centre \( O \) at a point \( Q \) so that \( OQ = 14 \text{ cm} \). The length of \( PQ \) is:
(a) \( 4\sqrt{10} \text{ cm} \)
(b) \( 6\sqrt{10} \text{ cm} \)
(c) \( 5\sqrt{10} \text{ cm} \)
(d) \( 7\sqrt{10} \text{ cm} \)
Answer: A

Question. Tangents \( AP \) and \( AQ \) are drawn to circle with centre \( O \) from an external point \( A \), then \( \angle PAQ \) is equal to:
(a) \( 2\angle OPQ \)
(b) \( \frac{\angle OPQ}{2} \)
(c) \( \frac{\angle OPQ}{3} \)
(d) \( \frac{\angle OPQ}{4} \)
Answer: A

Question. A circle of radius 2 lies in the first quadrant and touches both the axes of coordinates. The equation of the circle with centre at (6, 5) and touching the above circle externally is
(a) \( x^2 + y^2 + 12x - 10y + 52 = 0 \)
(b) \( x^2 + y^2 - 12x + 10y + 52 = 0 \)
(c) \( x^2 + y^2 - 12x - 10y + 52 = 0 \)
(d) None of these
Answer: C
Given, \( AC = 2 \), \( A \equiv (2, 2) \). Let, \( B \equiv (6, 5) \).
\( AB = \sqrt{(2-6)^2 + (2-5)^2} = 5 \)
\( BC = AB - AC = 5 - 2 = 3 \)

FILL IN THE BLANK

Question. The lengths of the two tangents from an external point to a circle are ..........
Answer: parallel

Question. A line that intersects a circle in one point only is called ..........
Answer: tangent

Question. The tangents drawn at the ends of a diameter of a circle are ..........
Answer: two

Question. A tangent of a circle touches it at .......... point(s).
Answer: one

Question. Tangent is perpendicular to the .......... through the point of contact.
Answer: radius

Question. A line intersecting a circle at two points is called a ..........
Answer: secant

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

TRUE/FALSE

Question. The tangent to the circumcircle of an isosceles triangle \( ABC \) at \( A \), in which \( AB = AC \), is parallel to \( BC \).
Answer: True

Question. A line drawn from the centre of a circle to a chord always bisects it.
Answer: False

Question. If angle between two tangents drawn from a point \( P \) to a circle of radius \( a \) and centre \( O \) is \( 90^{\circ} \), then \( OP = a\sqrt{2} \).
Answer: True

Question. Line joining the centers of two intersecting circles always bisect their common chord.
Answer: True

Question. In a circle, two chords \( PQ \) and \( RS \) bisect each other. Then \( PRQS \) is a rectangles.
Answer: True

Question. A tangent to a circle is a line that intersects the circle in only one point.
Answer: True

Question. The length of tangent from an external point \( P \) on a circle with centre \( O \) is always less than \( OP \).
Answer: True

Question. The angle between two tangents to a circle may be \( 0^{\circ} \).
Answer: False

Question. The tangent to a circle is a special case of the secant.
Answer: True

Question. If angle between two tangents drawn from a point \( P \) to a circle of radius \( a \) and centre \( O \) is \( 90^{\circ} \), then \( OP = a\sqrt{3} \).
Answer: False

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