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A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position is
- a)7πR/12
- b)1/2
- c)3/1
- d)NONE
Correct answer is option 'A'. Can you explain this answer?
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A point O is situated on a circle of radius R and with centre O anothe...
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To find the length of the arc described by the center of the smaller circle in moving from one extreme position to the other, we need to find the angle subtended by this arc at the center of the larger circle.
Let's call the center of the larger circle A and the center of the smaller circle B. Let's also call the points where the smaller circle is in contact with the larger circle C and D, with C being the starting position and D being the ending position.
Since AB is the radius of the larger circle and BC is the radius of the smaller circle, we have AB = R and BC = R/8.
Using the properties of circles and tangents, we can determine that triangle ABC is a right triangle. This is because AB is perpendicular to the tangent line at point C.
Let's call the point of tangency E. Triangle ABC is similar to triangle ABE, so we can use the ratios of corresponding sides to find the length of AE.
AE/AB = BE/BC
AE/R = (3R/2 - R)/(R/8)
AE/R = (R/2)/(R/8)
AE/R = 4
Therefore, AE = 4R.
Since triangle ABC is a right triangle, we can use the Pythagorean theorem to find the length of AC.
AC^2 = AB^2 + BC^2
AC^2 = R^2 + (R/8)^2
AC^2 = R^2 + R^2/64
AC^2 = (65/64)R^2
Therefore, AC = sqrt((65/64)R^2) = (sqrt(65)/8)R.
Now, we can find the angle subtended by the arc CD at point A using the definition of angle subtended by an arc on a circle.
Angle CAD = 2 * arcsin((AC/2)/(AB))
Angle CAD = 2 * arcsin(((sqrt(65)/8)R/2)/R)
Angle CAD = 2 * arcsin(sqrt(65)/16)
Finally, we can find the length of the arc CD using the definition of arc length on a circle.
Length of arc CD = R * Angle CAD
Length of arc CD = R * 2 * arcsin(sqrt(65)/16)
Therefore, the length of the arc described by the center of the smaller circle in moving from one extreme position to the other is 2R * arcsin(sqrt(65)/16).
Let's call the center of the larger circle A and the center of the smaller circle B. Let's also call the points where the smaller circle is in contact with the larger circle C and D, with C being the starting position and D being the ending position.
Since AB is the radius of the larger circle and BC is the radius of the smaller circle, we have AB = R and BC = R/8.
Using the properties of circles and tangents, we can determine that triangle ABC is a right triangle. This is because AB is perpendicular to the tangent line at point C.
Let's call the point of tangency E. Triangle ABC is similar to triangle ABE, so we can use the ratios of corresponding sides to find the length of AE.
AE/AB = BE/BC
AE/R = (3R/2 - R)/(R/8)
AE/R = (R/2)/(R/8)
AE/R = 4
Therefore, AE = 4R.
Since triangle ABC is a right triangle, we can use the Pythagorean theorem to find the length of AC.
AC^2 = AB^2 + BC^2
AC^2 = R^2 + (R/8)^2
AC^2 = R^2 + R^2/64
AC^2 = (65/64)R^2
Therefore, AC = sqrt((65/64)R^2) = (sqrt(65)/8)R.
Now, we can find the angle subtended by the arc CD at point A using the definition of angle subtended by an arc on a circle.
Angle CAD = 2 * arcsin((AC/2)/(AB))
Angle CAD = 2 * arcsin(((sqrt(65)/8)R/2)/R)
Angle CAD = 2 * arcsin(sqrt(65)/16)
Finally, we can find the length of the arc CD using the definition of arc length on a circle.
Length of arc CD = R * Angle CAD
Length of arc CD = R * 2 * arcsin(sqrt(65)/16)
Therefore, the length of the arc described by the center of the smaller circle in moving from one extreme position to the other is 2R * arcsin(sqrt(65)/16).
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A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position isa)7πR/12b)1/2c)3/1d)NONECorrect answer is option 'A'. Can you explain this answer?
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A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position isa)7πR/12b)1/2c)3/1d)NONECorrect answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position isa)7πR/12b)1/2c)3/1d)NONECorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position isa)7πR/12b)1/2c)3/1d)NONECorrect answer is option 'A'. Can you explain this answer?.
A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position isa)7πR/12b)1/2c)3/1d)NONECorrect answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position isa)7πR/12b)1/2c)3/1d)NONECorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A point O is situated on a circle of radius R and with centre O another circle of radius 3R/2 is described. Inside the smaller crescent-shaped region intercepted between these circles a circle of radius R/8 is placed. If smaller circle moves in contact with the circle of radius R, then the length of the arc described by its centre in moving from one extreme position to other extreme position isa)7πR/12b)1/2c)3/1d)NONECorrect answer is option 'A'. Can you explain this answer?.
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