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Consider the parametric equations below. x = t², y = t⁹. What type of curve is described by these equations?

a) Line
b) Circle
c) Parabola
d) Hyperbola

User Marcv
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Final answer:

The curve described by the equations x = t², y = t⁹ is more complex than a simple conic section and would exhibit a steep increase in y relative to x, especially for positive values of t. It's not a standard parabola, circle, line, or hyperbola.

Step-by-step explanation:

The curve described by the parametric equations x = t² and y = t⁹ is not a simple conic section like a line, circle, parabola, or hyperbola. For parametric equations that represent conic sections, the relationship between x and y is usually of a lower degree. In this case, because y is raised to the ninth power while x is squared, the curve will be more complex than the standard conic sections like those formed by the intersection of a plane with a cone, as stated in Figure 3.3. However, if we only focus on positive values of t, then as t increases, y will increase at a much faster rate than x, indicating that the curve will be very steep. For negative values of t, y will also be positive (because an odd power of a negative number is negative, but then it is squared to get y, making it positive again), but because t squared is always positive, x will also be positive and the curve will have symmetry with respect to the y-axis.

While this curve is related to a parabola because it has a squared term, it is not a parabola itself. The term 'parabolic' often refers to the specific equation of the form y = ax + bx², which is derived from projectile motion equations. When we solve for time t in the equation x = Voxt and substitute it into the expression for y = Voyt - (1/2)gt², we typically get an equation of this form, thus showing the parabolic nature of projectile trajectories.

User Kush Patel
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