Textbook Question
Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).
b. Explain why tan θ = y/x.
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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 12.1.95b
Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.
b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t
Verified step by step guidance1
Identify the parametric equations of the two lines: Line 1 is given by \(x = 2 + 5s\), \(y = 1 + s\) and Line 2 is given by \(x = 4 + 10t\), \(y = 3 + 2t\), where \(s\) and \(t\) are parameters.
Check if the lines are parallel by comparing their direction vectors. The direction vector of Line 1 is \(\langle 5, 1 \rangle\) and for Line 2 it is \(\langle 10, 2 \rangle\). Determine if one vector is a scalar multiple of the other.
If the direction vectors are scalar multiples, the lines are parallel. If not, they intersect. Since the direction vectors here appear related, verify by checking if \(\langle 10, 2 \rangle = k \langle 5, 1 \rangle\) for some scalar \(k\).
If the lines intersect, find the point of intersection by setting the parametric equations equal: \(2 + 5s = 4 + 10t\) and \(1 + s = 3 + 2t\). Solve this system of equations for \(s\) and \(t\).
Once you find values for \(s\) and \(t\), substitute back into either parametric equation to find the coordinates \((x, y)\) of the intersection point.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations of Lines
Parametric equations express the coordinates of points on a line as functions of a parameter, typically denoted by variables like s or t. Each parameter value corresponds to a unique point on the line, allowing a clear representation of lines in the plane or space.
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08:02
Parameterizing Equations
Determining Parallelism of Lines
Two lines are parallel if their direction vectors are scalar multiples of each other. By comparing the coefficients of the parameters in the parametric equations, one can check if the lines have the same direction, indicating parallelism.
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Finding the Point of Intersection
If lines are not parallel, their point of intersection can be found by equating their parametric expressions and solving the resulting system of equations for the parameters. Substituting these parameter values back into the equations gives the coordinates of the intersection point.
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Related Practice
Textbook Question
11–14. Working with parametric equations Consider the following parametric equations.
c. Eliminate the parameter to obtain an equation in x and y.
d. Describe the curve.
x=−t+6, y=3t−3; −5≤t≤5
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².
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Textbook Question
Spiral arc length Consider the spiral r=4θ, for θ≥0.
c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.
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Textbook Question
67–72. Derivatives Consider the following parametric curves.
b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t.
x = 2 + 4t, y = 4 − 8t; t = 2
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.
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