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The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given by
- a)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0
- b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0
- c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given by...
The hyperbola is given by
2x2 + 5xy + 2y2 + 4x + 5y = 0 ............................(i)
Since the equation of hyperbola will differ from equation of asymptote by a constant. So, equation of asymptote is
2x2 + 5xy + 2y3 + 4x + 5y + λ = 0 ......................(ii)
If (ii) represents 2 straight lines, we must have
abc + 2fgh - af2- bg2 - ch2 = 0
or 22λ + 2.5 / 2.4 / 2.5 / 2 - 2.25 / 4 - 2.4 - λ (25 / 4) = 0
or 9λ = 18 ∴ λ = 2
2x2 + 5xy + 2y3 + 4x + 5y + 2 = 0
2x2 + 5xy + 2y2 + 4x + 5y = 0 ............................(i)
Since the equation of hyperbola will differ from equation of asymptote by a constant. So, equation of asymptote is
2x2 + 5xy + 2y3 + 4x + 5y + λ = 0 ......................(ii)
If (ii) represents 2 straight lines, we must have
abc + 2fgh - af2- bg2 - ch2 = 0
or 22λ + 2.5 / 2.4 / 2.5 / 2 - 2.25 / 4 - 2.4 - λ (25 / 4) = 0
or 9λ = 18 ∴ λ = 2
2x2 + 5xy + 2y3 + 4x + 5y + 2 = 0
Most Upvoted Answer
The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given by...
Understanding the Curve
The given equation of the curve is:
2x^2 + 5xy + 2y^2 + 4x + 5y = 0.
This is a second-degree polynomial, and its general form indicates that it might represent a conic section. To find the asymptotes, we need to analyze the leading coefficients and their interactions.
Identifying the Asymptotes
To derive the asymptotes, we can consider the homogeneous part of the equation:
2x^2 + 5xy + 2y^2 = 0.
We can factor this equation to identify lines that describe the asymptotic behavior as x and y approach infinity.
Factoring the Homogeneous Part
To factor, we can treat the equation as a quadratic in y:
2y^2 + 5xy + 2x^2 = 0.
Using the quadratic formula, we can find the roots which represent the slopes of the asymptotes:
y = (-5x ± √(25x^2 - 16x^2))/(4).
This simplifies to:
y = (-5x ± 3x)/(4).
This leads to two asymptotes:
y = -x/2 and y = -2x.
Deriving the Asymptotic Equation
To get the explicit equation of the asymptotes, we need to eliminate the constant term from the original equation. The asymptotes can be represented by the modified equation:
2x^2 + 5xy + 2y^2 + 4x + 5y + 2 = 0.
This is option 'A', indicating that the asymptotes are represented by this equation, confirming the correct answer.
Conclusion
Thus, the curve's asymptotes are effectively captured by option 'A', providing a structured understanding of its behavior at infinity.
The given equation of the curve is:
2x^2 + 5xy + 2y^2 + 4x + 5y = 0.
This is a second-degree polynomial, and its general form indicates that it might represent a conic section. To find the asymptotes, we need to analyze the leading coefficients and their interactions.
Identifying the Asymptotes
To derive the asymptotes, we can consider the homogeneous part of the equation:
2x^2 + 5xy + 2y^2 = 0.
We can factor this equation to identify lines that describe the asymptotic behavior as x and y approach infinity.
Factoring the Homogeneous Part
To factor, we can treat the equation as a quadratic in y:
2y^2 + 5xy + 2x^2 = 0.
Using the quadratic formula, we can find the roots which represent the slopes of the asymptotes:
y = (-5x ± √(25x^2 - 16x^2))/(4).
This simplifies to:
y = (-5x ± 3x)/(4).
This leads to two asymptotes:
y = -x/2 and y = -2x.
Deriving the Asymptotic Equation
To get the explicit equation of the asymptotes, we need to eliminate the constant term from the original equation. The asymptotes can be represented by the modified equation:
2x^2 + 5xy + 2y^2 + 4x + 5y + 2 = 0.
This is option 'A', indicating that the asymptotes are represented by this equation, confirming the correct answer.
Conclusion
Thus, the curve's asymptotes are effectively captured by option 'A', providing a structured understanding of its behavior at infinity.
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The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given bya)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0d)None of these Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given bya)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0d)None of these Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given bya)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0d)None of these Correct answer is option 'A'. Can you explain this answer?.
The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given bya)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0d)None of these Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given bya)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0d)None of these Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given bya)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0d)None of these Correct answer is option 'A'. Can you explain this answer?.
Solutions for The asymptotes of the curve 2x2 + 5xy + 2y2 + 4x + 5y = 0 are given bya)2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0b)2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0c)2x2 + 5xy + 2y2 + 4x + 5y - 4 = 0d)None of these Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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