Welcome to Part 1 of our new 8-part series on Chapter 10, Circles. This post contains the top 25 Multiple Choice Questions (MCQs) to help you master the properties of tangents to a circle.
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Top 25 MCQs - Circles
Question 1: Tangent Length
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the center O at a point Q so that OQ = 12 cm. Length PQ is:
In right ΔOPQ, OQ² = OP² + PQ².
12² = 5² + PQ².
144 = 25 + PQ².
PQ² = 144 - 25 = 119.
PQ = √119 cm.
Question 2: Angle between Tangents
If tangents PA and PB from a point P to a circle with center O are inclined to each other at an angle of 80°, then ∠POA is equal to:
In quadrilateral OAPB, sum of angles is 360°. ∠OAP = ∠OBP = 90°.
∠AOB = 360 - (90+90+80) = 100°.
In ΔOAP and ΔOBP, they are congruent. So ∠POA = (1/2)∠AOB = 50°.
Question 3: Tangents from Internal Point
The number of tangents that can be drawn from a point inside a circle is:
Question 4: Distance from Center
The length of a tangent from a point A at distance 5 cm from the center of the circle is 4 cm. The radius of the circle is:
r² + 4² = 5² (Pythagoras theorem in right triangle formed by radius, tangent, and line joining center).
r² + 16 = 25 => r² = 9 => r = 3 cm.
Question 5: Parallel Tangents
The maximum number of parallel tangents a circle can have is:
Question 6: Angle at Center
If angle between two radii of a circle is 130°, the angle between the tangents at the ends of the radii is:
Angle between tangents + Angle between radii = 180°.
x + 130° = 180° => x = 50°.
Question 7: Quadrilateral Circumscribing
A quadrilateral ABCD circumscribes a circle. If AB = 6 cm, BC = 9 cm and CD = 8 cm, then the length of AD is:
6 + 8 = AD + 9
14 = AD + 9
AD = 14 - 9 = 5 cm.
Question 8: Concentric Circles
Two concentric circles are of radii 5 cm and 3 cm. The length of the chord of the larger circle which touches the smaller circle is:
In the right triangle formed by the radius of larger circle (5), radius of smaller circle (3), and half-chord:
Half-chord² = 5² - 3² = 25 - 9 = 16.
Half-chord = 4 cm.
Total length = 2 * 4 = 8 cm.
Question 9: Tangent Contact Point
At point A on a diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY at a distance 16 cm from A is:
Half chord = √(r² - d²) = √(10² - 6²) = √(100-36) = √64 = 8.
Total length = 16 cm.
Question 10: Secant vs Tangent
A line intersecting a circle in two points is called a:
Question 11: Perpendicular Tangent
The tangent at any point of a circle is perpendicular to the radius through the point of contact. This angle is:
Question 12: Number of Tangents
From a point P outside a circle, how many tangents can be drawn to the circle?
Question 13: Parallelogram Circumscribing
A parallelogram circumscribing a circle is a:
Question 14: Distance between Parallel Tangents
The distance between two parallel tangents to a circle of radius 5 cm is:
Diameter = 2 * Radius = 2 * 5 = 10 cm.
Question 15: Point on Circle
How many tangents can be drawn to a circle at a point on the circle?
Question 16: Equal Tangents
If PA and PB are tangents to a circle from an external point P, then:
Question 17: Quadrilateral Angles
A quadrilateral PQRS subtends angles at the center of the circle it circumscribes. If ∠POQ = 110°, then ∠ROS is:
∠POQ + ∠ROS = 180°.
110° + ∠ROS = 180° => ∠ROS = 70°.
Question 18: Length AB
In a figure, PA and PB are tangents to a circle with center O. If ∠APB = 60°, then length of chord AB is (given PA = 5 cm):
Since ∠APB = 60°, the other two angles are (180-60)/2 = 60°.
Thus, ΔPAB is equilateral. So, AB = PA = 5 cm.
Question 19: Common Tangents
Two circles touch each other externally at C. AB is a common tangent touching the circles at A and B. Then ∠ACB is:
Question 20: Perpendicular to Radius
A line drawn through the end point of a radius and perpendicular to it is a ______ to the circle.
Question 21: Angle OPA
If TP and TQ are tangents to a circle with center O such that ∠POQ = 110°, then ∠PTQ is equal to:
So, ∠POQ + ∠PTQ = 180°.
110° + ∠PTQ = 180° => ∠PTQ = 70°.
Question 22: Isosceles Tangents
In a circle of radius 7 cm, a tangent PT is drawn from a point P such that PT = 24 cm. If O is the center, then the length OP is:
OP² = 7² + 24².
OP² = 49 + 576 = 625.
OP = 25 cm.
Question 23: Non-Intersecting
Two circles of radii 5 cm and 3 cm touch each other internally. The distance between their centers is:
Distance = 5 - 3 = 2 cm.
Question 24: Perimeter of Triangle
A circle is inscribed in a triangle ABC touching AB, BC, and AC at P, Q, and R respectively. If AR = 5 cm, BQ = 6 cm, and CP = 8 cm (Note: Wait, CP should be CR or something. Let's assume standard tangents: AR=AP, BQ=BP, CQ=CR). If AR=5, BQ=6, CQ=8, then perimeter of ΔABC is:
AP = AR = 5
BP = BQ = 6
CR = CQ = 8
Perimeter = (AP+BP) + (BQ+CQ) + (CR+AR) = (5+6) + (6+8) + (8+5) = 11 + 14 + 13 = 38 cm.
Question 25: Chord & Concentric
If the radii of two concentric circles are 4 cm and 5 cm, then the length of each chord of one circle which is tangent to the other circle is:
Total length = 2 * 3 = 6 cm.
