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The equation of a curve whose slope at any point is thrice its abscissa and which passes through (–1, –3) is 2y = λ (x2 − 3)2y = λ(x2 − 3), then the value of λ must be
Correct answer is '3'. Can you explain this answer?
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Explanation:
Given information:
- The slope at any point is thrice its abscissa
- The curve passes through (-1, -3)
Equation of the curve:
Let the equation of the curve be y = λ(x^2 - 3)
Finding the slope:
Given that the slope at any point is thrice its abscissa, we have:
dy/dx = 3x
Equating the slopes:
Since the given curve has slope 3x, we have:
dy/dx = 3x = d/dx(λ(x^2 - 3))
Finding the derivative:
d/dx(λ(x^2 - 3)) = 2λx
Equating the derivatives:
3x = 2λx
Substitute the point (-1, -3):
-3 = 2λ(-1)
-3 = -2λ
λ = 3
Therefore, the value of λ must be 3.
Given information:
- The slope at any point is thrice its abscissa
- The curve passes through (-1, -3)
Equation of the curve:
Let the equation of the curve be y = λ(x^2 - 3)
Finding the slope:
Given that the slope at any point is thrice its abscissa, we have:
dy/dx = 3x
Equating the slopes:
Since the given curve has slope 3x, we have:
dy/dx = 3x = d/dx(λ(x^2 - 3))
Finding the derivative:
d/dx(λ(x^2 - 3)) = 2λx
Equating the derivatives:
3x = 2λx
Substitute the point (-1, -3):
-3 = 2λ(-1)
-3 = -2λ
λ = 3
Therefore, the value of λ must be 3.
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The equation of a curve whose slope at any point is thrice its abscissa and which passes through (–1, –3) is2y = λ (x2 − 3)2y = λ(x2 − 3), then the value ofλmust beCorrect answer is '3'. Can you explain this answer?
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