Textbook Question
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.4.74
Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)
Verified step by step guidance1
Start with the given hyperbola equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Implicitly differentiate both sides of the equation with respect to \(x\) to find \(\frac{dy}{dx}\). Use the chain rule for the \(y^2\) term, treating \(y\) as a function of \(x\).
After differentiating, you will get: \(\frac{2x}{a^2} + \frac{2y}{b^2} \cdot \frac{dy}{dx} = 0\). Solve this equation for \(\frac{dy}{dx}\) to find the slope of the tangent line at any point \((x, y)\) on the hyperbola.
Substitute the given point \((x_0, y_0)\) into the expression for \(\frac{dy}{dx}\) to find the slope of the tangent line at that specific point.
Use the point-slope form of a line, \(y - y_0 = m(x - x_0)\), where \(m\) is the slope found in the previous step, to write the equation of the tangent line at \((x_0, y_0)\).
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5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is used to find the derivative of y with respect to x when y is defined implicitly by an equation involving both variables. For the hyperbola equation, it allows us to differentiate both sides with respect to x to find the slope of the tangent line at a given point.
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Finding The Implicit Derivative
Equation of a Tangent Line
The tangent line to a curve at a point has a slope equal to the derivative of the curve at that point. Once the slope is found, the tangent line equation can be written using the point-slope form: y - y₀ = m(x - x₀), where m is the slope and (x₀, y₀) is the point of tangency.
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Hyperbola Equation and Properties
A hyperbola is defined by the equation x²/a² - y²/b² = 1 (or similar forms). Understanding its standard form and properties helps identify the correct implicit differentiation steps and interpret the geometric meaning of the tangent line at a specific point on the curve.
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Related Practice
Textbook Question
What is the polar equation of the horizontal line y = 5?
Textbook Question
15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x = r − 1, y = r³; −4 ≤ r ≤ 4
Textbook Question
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the curve r = √(cos θ)
Textbook Question
53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.
r = 1 - cos θ
1
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Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete cardioid r = 4 + 4 sin θ