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Two runners A, B starts simultaneously from the two points where the circle x2 + y2 = 16 cut the X-axis. The speeds of A, B is in the ratio 2: 1 and they are moving in the opposite directions.The coordinates of the point where they meet for the first time is
- a)(√3√2 , 1)
- b)(2, √3)
- c)(2, 2√3)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Two runners A, B starts simultaneously from the two points where the c...
Distance covered by A = (2/3) X (2π x 4/2) = 8π/3
Point C is such that angle COB = 60º, coordinates of C = (2, 2√3)
Since the angle is 60 degrees we can find coordinates by using 30 - 60 - 90 triangle, The sides are in ratio 1:√3:2 , If hypotenuse is 4, Then x axis is 4/2 = 2 and y axis is 4*√3/2 = 2√3
Most Upvoted Answer
Two runners A, B starts simultaneously from the two points where the c...
Approach:
We will first find the coordinates of runners A and B based on their speeds and starting points. Then we will calculate the time taken for them to meet for the first time and find the point of intersection.
Finding Coordinates of Runners A and B:
- Let the coordinates of A be (x1, 0) and the coordinates of B be (x2, 0).
- Since A and B start at the points where the circle x^2 + y^2 = 16 cuts the X-axis, we have x1 = -x2.
- Let the speeds of A and B be 2s and s, respectively. Thus, the coordinates of A and B will be (-2s*t, 0) and (s*t, 0) after time t.
Calculating Time Taken for First Meeting:
- The distance between A and B after time t is given by: 4s*t.
- The total distance covered by A and B when they meet is equal to the circumference of the circle which is 2*pi*4 = 8*pi.
- Therefore, 4s*t = 8*pi, giving t = 2*pi/s.
Calculating Point of Intersection:
- Substituting the value of t in the coordinates of A and B, we get A as (-4*pi, 0) and B as (2*pi, 0).
- The point of intersection is the average of the coordinates of A and B, which is ((-4*pi + 2*pi)/2, 0) = (-pi, 0).
- The x-coordinate of the point of intersection is -pi, which equals -3.14 approximately.
- Since the point lies on the circle x^2 + y^2 = 16, the y-coordinate can be calculated as y = sqrt(16 - (-3.14)^2) = sqrt(16 - 9.8596) = sqrt(6.1404) = 2.48 (approximately).
- Therefore, the point of intersection is approximately (-3.14, 2.48), which is closest to option C, (2, 2*sqrt(3)).
We will first find the coordinates of runners A and B based on their speeds and starting points. Then we will calculate the time taken for them to meet for the first time and find the point of intersection.
Finding Coordinates of Runners A and B:
- Let the coordinates of A be (x1, 0) and the coordinates of B be (x2, 0).
- Since A and B start at the points where the circle x^2 + y^2 = 16 cuts the X-axis, we have x1 = -x2.
- Let the speeds of A and B be 2s and s, respectively. Thus, the coordinates of A and B will be (-2s*t, 0) and (s*t, 0) after time t.
Calculating Time Taken for First Meeting:
- The distance between A and B after time t is given by: 4s*t.
- The total distance covered by A and B when they meet is equal to the circumference of the circle which is 2*pi*4 = 8*pi.
- Therefore, 4s*t = 8*pi, giving t = 2*pi/s.
Calculating Point of Intersection:
- Substituting the value of t in the coordinates of A and B, we get A as (-4*pi, 0) and B as (2*pi, 0).
- The point of intersection is the average of the coordinates of A and B, which is ((-4*pi + 2*pi)/2, 0) = (-pi, 0).
- The x-coordinate of the point of intersection is -pi, which equals -3.14 approximately.
- Since the point lies on the circle x^2 + y^2 = 16, the y-coordinate can be calculated as y = sqrt(16 - (-3.14)^2) = sqrt(16 - 9.8596) = sqrt(6.1404) = 2.48 (approximately).
- Therefore, the point of intersection is approximately (-3.14, 2.48), which is closest to option C, (2, 2*sqrt(3)).
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Two runners A, B starts simultaneously from the two points where the circle x2 + y2 = 16 cut the X-axis. The speeds of A, B is in the ratio 2: 1 and they are moving in the opposite directions.The coordinates of the point where they meet for the first time isa)(√3√2 , 1)b)(2, √3)c)(2, 2√3)d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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Two runners A, B starts simultaneously from the two points where the circle x2 + y2 = 16 cut the X-axis. The speeds of A, B is in the ratio 2: 1 and they are moving in the opposite directions.The coordinates of the point where they meet for the first time isa)(√3√2 , 1)b)(2, √3)c)(2, 2√3)d)None of theseCorrect 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 runners A, B starts simultaneously from the two points where the circle x2 + y2 = 16 cut the X-axis. The speeds of A, B is in the ratio 2: 1 and they are moving in the opposite directions.The coordinates of the point where they meet for the first time isa)(√3√2 , 1)b)(2, √3)c)(2, 2√3)d)None of theseCorrect 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 runners A, B starts simultaneously from the two points where the circle x2 + y2 = 16 cut the X-axis. The speeds of A, B is in the ratio 2: 1 and they are moving in the opposite directions.The coordinates of the point where they meet for the first time isa)(√3√2 , 1)b)(2, √3)c)(2, 2√3)d)None of theseCorrect answer is option 'C'. Can you explain this answer?.
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