Textbook Question
General equations for a circle Prove that the equations
X = a cos t + b sin t, y = c cos t + d sin t
where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0.
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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.R.1e
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.
Verified step by step guidance1
Recall that the x-intercepts of a curve occur where the graph crosses the x-axis, which means the y-coordinate is zero. So, to find the x-intercepts, set \(y = 0\) in the equation of the hyperbola.
Substitute \(y = 0\) into the equation \(\frac{y^2}{2} - \frac{x^2}{4} = 1\), which simplifies to \(\frac{0^2}{2} - \frac{x^2}{4} = 1\), or \(-\frac{x^2}{4} = 1\).
Multiply both sides of the equation by \(-4\) to isolate \(x^2\): \(x^2 = -4\).
Since \(x^2 = -4\) has no real solutions (because the square of a real number cannot be negative), there are no real values of \(x\) that satisfy the equation when \(y=0\).
Therefore, the hyperbola \(\frac{y^2}{2} - \frac{x^2}{4} = 1\) has no x-intercepts, confirming the statement is true.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of x-intercept
An x-intercept is a point where a graph crosses the x-axis, meaning the y-coordinate is zero. To find x-intercepts, set y = 0 in the equation and solve for x. If no real solutions exist, the graph has no x-intercepts.
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Definition of the Definite Integral
Equation of a hyperbola
A hyperbola is a conic section defined by an equation of the form (y²/a²) - (x²/b²) = 1 or (x²/a²) - (y²/b²) = 1. It represents two separate curves and its shape depends on the signs and values of a² and b².
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Solving for intercepts in conic sections
To determine intercepts in conic sections, substitute the corresponding coordinate (x=0 for y-intercept, y=0 for x-intercept) into the equation. Analyze the resulting equation for real solutions to confirm the existence of intercepts.
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Related Practice
Textbook Question
14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The circle x ² + y ² =9, generated clockwise
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Textbook Question
58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.
x²/16 - y²/9 = 1; (20/3, -4)
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Textbook Question
53–57. Conic sections
d. Make an accurate graph of the curve.
x = 16y²
Textbook Question
7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3
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Textbook Question
Polar conversion Consider the equation r=4/(sinθ+cosθ).
a. Convert the equation to Cartesian coordinates and identify the curve it describes.
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