Domanda del libro di testo
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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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 12, Problema 12.1.95c
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.
c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t
Guida verificata passo dopo passo1
Identify the parametric equations of the two lines: Line 1 is given by \(x = 1 + 3s\), \(y = 4 + 2s\) and Line 2 is given by \(x = 4 - 3t\), \(y = 6 + 4t\), where \(s\) and \(t\) are parameters.
To check if the lines are parallel, compare their direction vectors. The direction vector of Line 1 is \(\langle 3, 2 \rangle\) and for Line 2 it is \(\langle -3, 4 \rangle\). Determine if one vector is a scalar multiple of the other.
If the direction vectors are not scalar multiples, the lines are not parallel and may intersect. To find the intersection point, set the \(x\) and \(y\) coordinates equal to each other: \(1 + 3s = 4 - 3t\) and \(4 + 2s = 6 + 4t\).
Solve the system of equations for \(s\) and \(t\). This involves rearranging the equations and using substitution or elimination methods to find values of \(s\) and \(t\) that satisfy both equations simultaneously.
Once you find \(s\) and \(t\), substitute either value back into the parametric equations of one of the lines to find the coordinates of the intersection point \(\left(x, y\right)\).
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Parametric Equations of Lines
Parametric equations express the coordinates of points on a line as functions of a parameter, typically denoted by 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 Intersection of Lines
To find if two lines intersect, set their parametric equations equal and solve for the parameters. If a consistent solution exists, the lines intersect at the corresponding point; otherwise, they are parallel or skew.
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Percorso guidato
05:13Slopes of Tangent Lines
Parallel Lines and Direction Vectors
Two lines are parallel if their direction vectors are scalar multiples of each other. Comparing the coefficients of the parameters in the parametric equations helps identify parallelism without solving for intersection.
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Percorso guidato
05:13Slopes of Tangent Lines
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c. Eliminate the parameter to obtain an equation in x and y.
d. Describe the curve.
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