The pair of parametric equations represent a line, parabola, circle, ellipse, or hyperbola. Name the type of basic curve that the pair of equations represents.
[tex]
\begin{array}{l}
x+4=-6 t \\
y-8=4 t
\end{array}
[/tex]
Answer (2)
Solve the first equation for t : t = − 6 x + 4 .
Solve the second equation for t : t = 4 y − 8 .
Set the two expressions for t equal to each other: − 6 x + 4 = 4 y − 8 .
Simplify the equation to get the equation of a line: 2 x + 3 y = 16 . The answer is l in e .
Explanation
Problem Analysis We are given a pair of parametric equations:
x + 4 = − 6 t y − 8 = 4 t
Our goal is to determine what type of basic curve these equations represent. The options are line, parabola, circle, ellipse, or hyperbola.
Solving for t in terms of x To identify the curve, we need to eliminate the parameter t and find a direct relationship between x and y .
From the first equation, we can express t in terms of x :
t = − 6 x + 4
Solving for t in terms of y From the second equation, we can express t in terms of y :
t = 4 y − 8
Equating the expressions for t Since both expressions are equal to t , we can set them equal to each other:
− 6 x + 4 = 4 y − 8
Eliminating the fractions Now, we simplify the equation to eliminate the fractions. Multiply both sides by -12:
− 12 × − 6 x + 4 = − 12 × 4 y − 8
2 ( x + 4 ) = − 3 ( y − 8 )
Simplifying the equation Expand and rearrange the equation:
2 x + 8 = − 3 y + 24
2 x + 3 y = 24 − 8
2 x + 3 y = 16
Identifying the curve The resulting equation is of the form A x + B y = C , where A = 2 , B = 3 , and C = 16 . This is the equation of a line.
Final Answer Therefore, the pair of parametric equations represents a line.
Examples
Parametric equations are useful in computer graphics to draw curves and lines. For example, if you are designing a video game, you can use parametric equations to define the trajectory of a projectile or the path of a moving character. In this case, the equations describe a straight line, which could represent a simple movement pattern.
The given parametric equations represent a line. We eliminate the parameter t and obtain a linear equation in standard form, confirming it describes a straight line. Therefore, the correct answer is a line.
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