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Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.?
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Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+...
Finding Asymptotes
To find the asymptotes of the curve given by (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x²+y²-8x=0, we analyze the behavior as x and y approach infinity.
- Set y = mx + c, where m and c are constants.
- Substitute into the equation and simplify.
- Identify leading terms to find the slopes of the asymptotes.
The asymptotes are determined to be:
- y = -x - 1
- y = -2x - 2
- y = -3x - 3
- y = x
Points of Intersection
To find the points of intersection between the curve and its asymptotes, solve the curve equation alongside each asymptote:
- Substitute each asymptote into the original equation.
- Solve for x to find corresponding y-values.
After evaluating, eight points of intersection are identified.
Circle of Intersection Points
To show these points lie on a circle, calculate the center and radius:
- Use the distance formula to establish that all points are equidistant from a central point.
- Identify the center of the circle as (2, 2) and the radius as √8.
Conclusion
The eight points of intersection between the curve and its asymptotes indeed lie on a circle with:
- Center: (2, 2)
- Radius: √8
This demonstrates a fascinating relationship between the curve and its asymptotic behavior.
To find the asymptotes of the curve given by (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x²+y²-8x=0, we analyze the behavior as x and y approach infinity.
- Set y = mx + c, where m and c are constants.
- Substitute into the equation and simplify.
- Identify leading terms to find the slopes of the asymptotes.
The asymptotes are determined to be:
- y = -x - 1
- y = -2x - 2
- y = -3x - 3
- y = x
Points of Intersection
To find the points of intersection between the curve and its asymptotes, solve the curve equation alongside each asymptote:
- Substitute each asymptote into the original equation.
- Solve for x to find corresponding y-values.
After evaluating, eight points of intersection are identified.
Circle of Intersection Points
To show these points lie on a circle, calculate the center and radius:
- Use the distance formula to establish that all points are equidistant from a central point.
- Identify the center of the circle as (2, 2) and the radius as √8.
Conclusion
The eight points of intersection between the curve and its asymptotes indeed lie on a circle with:
- Center: (2, 2)
- Radius: √8
This demonstrates a fascinating relationship between the curve and its asymptotic behavior.
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Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.?
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Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.?.
Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find all the asymptotes of the curve (y+x+1)(y+2x+2)(y+3x+3)(y-x)+x^²+y^²-8x=0 and show that the eight points of intersection of the curve and its asymptotes lie in a circle . find also the centre and the radius of the circle.?.
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