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Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x - 9y),
- a)15
- b)11
- c)16
- d)21
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maxi...
Given equation is
Let x - 2 = cos θ
Then 4x - 9y = 11+ 4 cosθ - 3sinθ
Let x - 2 = cos θ
Then 4x - 9y = 11+ 4 cosθ - 3sinθ
Most Upvoted Answer
Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maxi...
Understanding the Relation
The given equation is:
x² + 9y² - 4x + 6y + 4 = 0.
This represents an ellipse. We can rewrite it in standard form by completing the square.
Completing the Square
- For x:
x² - 4x can be rewritten as (x - 2)² - 4.
- For y:
9y² + 6y can be rewritten as 9(y² + (2/3)y). Completing the square gives:
9[(y + 1/3)² - 1/9] = 9(y + 1/3)² - 1.
Substituting back, we have:
(x - 2)² - 4 + 9(y + 1/3)² - 1 + 4 = 0, which simplifies to:
(x - 2)² + 9(y + 1/3)² = 1.
Finding the Maximum of 4x - 9y
We need to maximize the expression:
z = 4x - 9y.
Using the parametric form of the ellipse, we can express x and y in terms of a parameter θ:
x = 2 + cos(θ), y = -1/3 + (1/3)sin(θ).
Substituting these into z gives:
z = 4(2 + cos(θ)) - 9(-1/3 + (1/3)sin(θ)).
This simplifies to:
z = 8 + 4cos(θ) + 3 - 3sin(θ) = 11 + 4cos(θ) - 3sin(θ).
Maximizing the Expression
To maximize:
11 + 4cos(θ) - 3sin(θ), we can rewrite it as:
11 + Rcos(θ + φ), where R = √(4² + (-3)²) = 5.
Thus, the maximum value of z is:
11 + 5 = 16.
Conclusion
The maximum value of the expression 4x - 9y is 16, confirming that option 'C' is correct.
The given equation is:
x² + 9y² - 4x + 6y + 4 = 0.
This represents an ellipse. We can rewrite it in standard form by completing the square.
Completing the Square
- For x:
x² - 4x can be rewritten as (x - 2)² - 4.
- For y:
9y² + 6y can be rewritten as 9(y² + (2/3)y). Completing the square gives:
9[(y + 1/3)² - 1/9] = 9(y + 1/3)² - 1.
Substituting back, we have:
(x - 2)² - 4 + 9(y + 1/3)² - 1 + 4 = 0, which simplifies to:
(x - 2)² + 9(y + 1/3)² = 1.
Finding the Maximum of 4x - 9y
We need to maximize the expression:
z = 4x - 9y.
Using the parametric form of the ellipse, we can express x and y in terms of a parameter θ:
x = 2 + cos(θ), y = -1/3 + (1/3)sin(θ).
Substituting these into z gives:
z = 4(2 + cos(θ)) - 9(-1/3 + (1/3)sin(θ)).
This simplifies to:
z = 8 + 4cos(θ) + 3 - 3sin(θ) = 11 + 4cos(θ) - 3sin(θ).
Maximizing the Expression
To maximize:
11 + 4cos(θ) - 3sin(θ), we can rewrite it as:
11 + Rcos(θ + φ), where R = √(4² + (-3)²) = 5.
Thus, the maximum value of z is:
11 + 5 = 16.
Conclusion
The maximum value of the expression 4x - 9y is 16, confirming that option 'C' is correct.
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Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x -9y),a)15b)11c)16d)21Correct answer is option 'C'. Can you explain this answer?
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Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x -9y),a)15b)11c)16d)21Correct answer is option 'C'. 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 Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x -9y),a)15b)11c)16d)21Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x -9y),a)15b)11c)16d)21Correct answer is option 'C'. Can you explain this answer?.
Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x -9y),a)15b)11c)16d)21Correct answer is option 'C'. 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 Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x -9y),a)15b)11c)16d)21Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let x and y satisfy the relation x2 + 9y2 - 4x + 6y + 4 = 0, then maximum value of the expression (4x -9y),a)15b)11c)16d)21Correct answer is option 'C'. Can you explain this answer?.
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