Question

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation.

Answer 1

To eliminate the parameter in parametric equations, we solve for x and y in terms of the parameter. The curve is described by the equation y = ax + bx^2, where the signs of a and b determine the orientation.

Step-by-step explanation:

To eliminate the parameter and obtain an equation in x and y, we can equate the two given parametric equations:

x = a + bt

y = c + dt

We can solve the first equation for a:

a = x - bt

Substituting the value of a into the second equation:

y = c + d(x - bt)

Expanding and rearranging terms:

y = c + dx - bdt

y = (c - bdt) + dx

So, the equation in x and y is y = (c - bdt) + dx.

For part b, the curve described by the equation will depend on the values of b, d, and c. If b > 0, the curve will slope upward to the right. If b = 0, the curve will be horizontal. If b < 0, the curve will slope downward to the right. The positive orientation of the curve will depend on the values of d and c in relation to the slope.