CAT Exam > CAT Questions > Two circles with radius 2R and √2R intersect...
Start Learning for Free
Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.
- a)(√3- π - 1) R2
- b)(√3 - π) R2
- c)(13π/6 + 1 - √3 ) R2
- d)(13π/6 +√3) R2
- e)None of the above
Correct answer is option 'C'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Two circles with radius 2R and √2R intersect each other at points A a...
Let us draw the diagram according to the given info,
View all questions of this test
We can see that AD = A0*cos60° = 2R* 1/2 = R
In triangle, ACD
=>sinACD = AC/AC
=>sinACD = R/√2 * R = 1/√2
=>∠ACD = 45°
By symmetry we can say that ∠BCD = 45°
Therefore we can say that ∠ACB = 90°
Hence, the area colored by green color = 270°/360° * π * (√2R)2 = 3/2 * π * R2... (1)
Area of triangle ACB = 1/2 * R* 2R = R2... (2)
Area shown in blue color = 60°/360° *π* (2R)2-√3/4 * (2R)2 = 2/3 *π* R2-√3 * R2... (3)
By adding (1) + (2) + (3)
Therefore, the area of the common region between two circles = 3/2 * π * R2 + R2 + 2/3 * π * R2 - √3 * R2
=> (13π/6 + 1 +√3)R2Hence, option C is the correct answer.
Most Upvoted Answer
Two circles with radius 2R and √2R intersect each other at points A a...
To find the area of the common region between the two circles, we can break it down into three parts: the sector of the larger circle, the sector of the smaller circle, and the triangle formed by the centers of the circles and point B.
1. Sector of the larger circle:
The angle at the center of the larger circle is 60° (given), so the angle at the circumference is also 60°. The area of the sector is given by (60/360) * π(2R)^2 = (1/6) * 4πR^2 = (2/3)πR^2.
2. Sector of the smaller circle:
The angle at the center of the smaller circle is also 60° (since the circles intersect at point B). The radius of the smaller circle is √2R, so the area of the sector is (1/6) * π(√2R)^2 = (1/6) * 2πR^2 = (1/3)πR^2.
3. Triangle formed by the centers and point B:
We can see that this triangle is an equilateral triangle, because the angles at the centers are equal (60°) and the sides are equal (2R and √2R). The area of an equilateral triangle with side length s is given by (√3/4) * s^2. In this case, the side length is 2R, so the area of the triangle is (√3/4) * (2R)^2 = (√3/4) * 4R^2 = √3R^2.
Therefore, the total area of the common region is the sum of the areas of the three parts:
(2/3)πR^2 + (1/3)πR^2 + √3R^2
= (3/3)πR^2 + √3R^2
= (π + √3)R^2
So, the correct answer is option C: (13π/6 - 1 - √3)R^2.
1. Sector of the larger circle:
The angle at the center of the larger circle is 60° (given), so the angle at the circumference is also 60°. The area of the sector is given by (60/360) * π(2R)^2 = (1/6) * 4πR^2 = (2/3)πR^2.
2. Sector of the smaller circle:
The angle at the center of the smaller circle is also 60° (since the circles intersect at point B). The radius of the smaller circle is √2R, so the area of the sector is (1/6) * π(√2R)^2 = (1/6) * 2πR^2 = (1/3)πR^2.
3. Triangle formed by the centers and point B:
We can see that this triangle is an equilateral triangle, because the angles at the centers are equal (60°) and the sides are equal (2R and √2R). The area of an equilateral triangle with side length s is given by (√3/4) * s^2. In this case, the side length is 2R, so the area of the triangle is (√3/4) * (2R)^2 = (√3/4) * 4R^2 = √3R^2.
Therefore, the total area of the common region is the sum of the areas of the three parts:
(2/3)πR^2 + (1/3)πR^2 + √3R^2
= (3/3)πR^2 + √3R^2
= (π + √3)R^2
So, the correct answer is option C: (13π/6 - 1 - √3)R^2.
Attention CAT Students!
To make sure you are not studying endlessly, EduRev has designed CAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in CAT.
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?
Question Description
Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?.
Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?.
Solutions for Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. 0 is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles.a)(√3- π - 1) R2b)(√3 - π) R2c)(13π/6 + 1 - √3 ) R2d)(13π/6 +√3) R2e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup