Finding Parametric Equations and Tangent Lines


Find parametric equations for the given curve.


9x² + 4y² = 36

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The circle x ² + y ² =9, generated clockwise

Air drop A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by

x = 80t, y = −4.9t² + 3000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release (see figure). Graph the trajectory of the packet and find the coordinates of the point where the packet lands.

Air drop—inverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by

x = 100t, y = −4.9t² + 4000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

{Use of Tech} Implicit function graph Explain and carry out a method for graphing the curve x = 1 + cos² y − sin² y using parametric equations and a graphing utility.

Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the line segment with endpoints (-1,3) and (3,-2)

Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the ray (half line) with initial point (-1,2) that passes through the point (0,0)

Finding Cartesian from Parametric Equations

In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.

x = cos t, y = sin 3t

Finding Parametric Equations

Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².

a. once clockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)