Theorem on Tangents MCQ Quiz in मल्याळम - Objective Question with Answer for Theorem on Tangents - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക
Last updated on Mar 11, 2025
Latest Theorem on Tangents MCQ Objective Questions
Top Theorem on Tangents MCQ Objective Questions
Theorem on Tangents Question 1:
In the given figure, chords XY and PQ intersect each other at point L. Find the length of XY (in cm).
Answer (Detailed Solution Below)
Theorem on Tangents Question 1 Detailed Solution
Calculation
By the theorem,
LQ × LP = LY × LX
Let the length of XY be x.
⇒ 5 × 15 = 3 × (3 + x)
⇒ 25 = x + 3
⇒ x = 22
The length of XY is 22.
Theorem on Tangents Question 2:
Two circles touch each other externally. The radius of the first circle with centre O is 6 cm. The radius of the second circle with centre P is 3 cm. Find the length of their common tangent AB.
Answer (Detailed Solution Below)
Theorem on Tangents Question 2 Detailed Solution
Given:
The radius of smaller circle = 3 cm
The radius of large circle = 6 cm
Formula used:
Direct common tangent = 2 × √(R × r)
Where, R = radius of large circle;
and r = radius of the small circle
Calculation:
Direct common tangent = 2 × √(R × r)
⇒ 2 × √(3 × 6)
⇒ 2 × 3 × √2 = 6√2 cm
∴ The correct answer is 6√2 cm.
Theorem on Tangents Question 3:
In the given figure, if PA and PB are tangents to the circle with centre O such that ∠APB = 54°, then ∠OBA = ________.
Answer (Detailed Solution Below)
Theorem on Tangents Question 3 Detailed Solution
Given:
PA and PB are tangents to the circle with center O such that ∠APB = 54°
Calculation:
We know that the radius and tangent are perpendicular at their point of contact.
So,
∠PAO = ∠PBO = 90°
So,
∠AOB = 360 - 54 - 180
⇒ 126°
Also AO = OB = Radius
So, ∠OAB = ∠OBA = 54/2
⇒ 27°
∴ The required answer is 27°
Theorem on Tangents Question 4:
If PT is a tangent at T to a circle whose centre is O and OP = 17 cm and OT = 8 cm, find the length of the tangent segment PT
Answer (Detailed Solution Below)
Theorem on Tangents Question 4 Detailed Solution
Given:
PT is a tangent to a circle, whose centre is O.
OP = 17 cm
OT = 8 cm
Formula used:
Pythagoras Theorem: In a right-angle triangle, the hypotenuse is equal to -
OP2 = OT2 + TP2
Calculation:
This, the length of the tangent TP is equal to:
TP2 = OP2 - OT2
TP2 = (17)2 - (8)2
TP2 = 289 - 64
TP2 = 225
TP = √225 = 15
∴ The length of the tangent segment PT is equal to 15 cm.
Theorem on Tangents Question 5:
The diameters of two circles are 12 cm and 20 cm, respectively and the distance between their centres is 16 cm. Find the number of common tangents to the circles.
Answer (Detailed Solution Below)
Theorem on Tangents Question 5 Detailed Solution
Given:
The diameters of the two circles are 12 cm and 20 cm
The distance between their centres is 16 cm
Calculation:
According to the diagram,
The two circles are touching each other externally
So, the number of common tangents will be 3
∴ The required answer is 3.
Theorem on Tangents Question 6:
Let C be a circle with centre O and P be an external point to C. Let PA and PB be two tangents to C with A and B being the points of tangency, respectively. If PA and PB are inclined to each other at an angle of 60º, then find ∠POA.
Answer (Detailed Solution Below)
Theorem on Tangents Question 6 Detailed Solution
Given:
∠APB = 60º
Formula used:
In ΔOAP and ΔOBP:
OA = OB (radii of the circle)
AP = BP (length of tangents from an external point)
OP = OP (common side)
By SSS congruence, ΔOAP ≅ ΔOBP
∠POA = ∠POB
∠OPA = ∠OPB
Therefore, OP is the angle bisector of ∠APB and ∠AOB
Hence, ∠OPA = ∠OPB = 1/2 (∠APB )
= 1/2 × 60°
= 30°
Calculation:
By angle sum property of a triangle,
In ΔOAP
∠A + ∠POA + ∠OPA = 180°
OA ⊥ AP (The tangent at any point of a circle is perpendicular to the radius through the point of contact.)
Therefore, ∠A = 90°
90° + ∠POA + 30° = 180°
120° + ∠POA = 180°
∠POA = 180° - 120°
∠POA = 60°
Therefore, Option (2) i.e 60° is the correct answer.
Theorem on Tangents Question 7:
P and Q are centres of two circles whose radii are 7 cm and 3 cm, respectively. If the direct common tangents to the circles meet PQ extended at A, then A divides PQ ________.
Answer (Detailed Solution Below)
Theorem on Tangents Question 7 Detailed Solution
Given:
Radius of circle with center P = 7 cm.
Radius of circle with center Q = 3 cm.
Formula Used:
Ratio of division of line segment by external tangents = Ratio of radii of circles
Calculation:
Let the direct common tangents to the circles meet PQ extended at A.
Since the tangents are direct common tangents, they will divide the line segment PQ externally in the ratio of the radii of the circles.
Ratio of radii of the circles = 7:3
Therefore, A divides PQ externally in the ratio 7:3
The correct answer is option 3.
Theorem on Tangents Question 8:
Find the value of x in the given figure where TF ( tangent on point T) = 15 cm, FB = 2x + 3 cm, FA = 9 cm.
Answer (Detailed Solution Below)
Theorem on Tangents Question 8 Detailed Solution
TF = 15 cm, FB = 2x + 3 cm and FA = 9 cm
Concept used:-
According to the tangent theorem,
TF2 = FA × FB
Calculation:-
TF2 = FA × FB
⇒ 225 = 9 × (2x + 3)
⇒ (2x + 3) = 25
⇒ x = 11cm
∴ The required result will be 11.
Theorem on Tangents Question 9:
The number of parallel tangents of a circle with a given tangent is:
Answer (Detailed Solution Below)
Theorem on Tangents Question 9 Detailed Solution
Concept used:
Explanation:
The largest chord in a circle is the diameter. This meets the circle at two points. The points of tangency lie on the two points where the diameter meets the circle.
∴ A circle can have 2 parallel tangents at most.
So, only one parallel tangent is possible for a given tangent.
Important points:
This diameter meets the circle in exactly two points. So there is no more space for a third or more parallel tangents
Theorem on Tangents Question 10:
What is the number of common tangents that can be drawn to two circles that touch each other externally?
Answer (Detailed Solution Below)
Theorem on Tangents Question 10 Detailed Solution
Calculation :
∴ The correct answer is 3.