Math Problem Statement
Use implicit differentiation to find the points where the parabola defined by x2 - 2xy + y2 + 2x - 6y + 21 = 0 has horizontal and vertical tangent lines. The parabola has horizontal tangent lines at the point(s) The parabola has vertical tangent lines at the point(s)
Solution
We are tasked with finding the points on the given parabola where there are horizontal and vertical tangent lines using implicit differentiation.
Given equation of the parabola:
Step 1: Implicit Differentiation
We differentiate both sides of the equation with respect to . Keep in mind that is a function of , so apply the chain rule where necessary.
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Differentiating :
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Differentiating :
Using the product rule:
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Differentiating :
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Differentiating :
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Differentiating :
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Differentiating the constant 21:
Step 2: Combine all differentiated terms
Simplify the equation:
Group the terms with and the rest:
Step 3: Solve for
Simplifying the expression:
Step 4: Conditions for Horizontal and Vertical Tangents
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Horizontal Tangent Line occurs when . Set the numerator equal to zero:
Simplifying:
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Vertical Tangent Line occurs when is undefined, i.e., when the denominator equals zero:
Simplifying:
Step 5: Solve for Points of Intersection
Now, we need to find the points where these lines intersect the original equation .
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For horizontal tangents ():
Substitute into the original equation:
Simplifying this:
Solving for :
Substituting into :
So the point where the parabola has a horizontal tangent is .
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For vertical tangents ():
Substitute into the original equation:
Simplifying this:
This leads to a contradiction, so there are no solutions for vertical tangents.
Final Answer:
- The parabola has horizontal tangent lines at the point .
- The parabola has no vertical tangent lines.
Would you like further clarification or additional questions on implicit differentiation? Here are some related questions:
- How does the chain rule affect implicit differentiation?
- What are the conditions
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Parabolas
Tangent Lines
Differential Calculus
Formulas
Implicit differentiation formula
Horizontal tangent condition (dy/dx = 0)
Vertical tangent condition (dy/dx undefined)
Theorems
Chain rule
Product rule
Quotient rule
Suitable Grade Level
Grades 11-12
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