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The curve represented by x = a (coshθ + sinhθ) , y = b(coshθ − sinhθ) is
- a)A hyperbola
- b)An ellipse
- c)A parabola
- d)A circle
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The curve represented by x = a (coshθ + sinhθ), y = b(cosh...
Understanding the Curve
The given parameterization of the curve is:
- x = a (coshθ + sinhθ)
- y = b (coshθ - sinhθ)
To determine the nature of the curve, we will manipulate these equations.
Transforming the Equations
1. Expressing Hyperbolic Functions:
- Recall the definitions:
- coshθ = (e^θ + e^(-θ)) / 2
- sinhθ = (e^θ - e^(-θ)) / 2
2. Substituting Definitions:
- Substitute coshθ and sinhθ into the equations for x and y:
- x = a [(e^θ + e^(-θ)) / 2 + (e^θ - e^(-θ)) / 2]
- y = b [(e^θ + e^(-θ)) / 2 - (e^θ - e^(-θ)) / 2]
3. Simplifying:
- After simplification, we obtain:
- x = a e^θ
- y = b e^(-θ)
Eliminating the Parameter
4. Relating x and y:
- From x = a e^θ, we have e^θ = x/a.
- From y = b e^(-θ), we get e^(-θ) = y/b.
- Multiplying these two equations yields:
- (x/a)(y/b) = 1
- Rearranging gives us:
- xy = ab
Conclusion: Hyperbola
5. Identifying the Conic Section:
- The equation xy = ab is a standard form of a hyperbola.
- Hyperbolas are characterized by the product of their coordinates being a constant (in this case, ab).
Thus, the curve represented by the given parameterization is indeed a hyperbola, confirming that the correct answer is option 'A'.
The given parameterization of the curve is:
- x = a (coshθ + sinhθ)
- y = b (coshθ - sinhθ)
To determine the nature of the curve, we will manipulate these equations.
Transforming the Equations
1. Expressing Hyperbolic Functions:
- Recall the definitions:
- coshθ = (e^θ + e^(-θ)) / 2
- sinhθ = (e^θ - e^(-θ)) / 2
2. Substituting Definitions:
- Substitute coshθ and sinhθ into the equations for x and y:
- x = a [(e^θ + e^(-θ)) / 2 + (e^θ - e^(-θ)) / 2]
- y = b [(e^θ + e^(-θ)) / 2 - (e^θ - e^(-θ)) / 2]
3. Simplifying:
- After simplification, we obtain:
- x = a e^θ
- y = b e^(-θ)
Eliminating the Parameter
4. Relating x and y:
- From x = a e^θ, we have e^θ = x/a.
- From y = b e^(-θ), we get e^(-θ) = y/b.
- Multiplying these two equations yields:
- (x/a)(y/b) = 1
- Rearranging gives us:
- xy = ab
Conclusion: Hyperbola
5. Identifying the Conic Section:
- The equation xy = ab is a standard form of a hyperbola.
- Hyperbolas are characterized by the product of their coordinates being a constant (in this case, ab).
Thus, the curve represented by the given parameterization is indeed a hyperbola, confirming that the correct answer is option 'A'.
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The curve represented by x = a (coshθ + sinhθ), y = b(coshθ − sinhθ)isa)A hyperbolab)An ellipsec)A parabolad)A circleCorrect answer is option 'A'. Can you explain this answer?
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