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.

Identifying Parametric Equations in the Plane

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π

Graph the plane curve formed by the parametric equations and indicate its orientation.

$x\left(t\right)=-t+1$; $y\left(t\right)=t^2$

$-2\le t\le 2$

Graph the plane curve formed by the parametric equations and indicate its orientation.

$x\left(t\right)=2t-1$; $y\left(t\right)=2\sqrt{t}$

$t\ge 0$

Eliminate the parameter to rewrite the following as a rectangular equation.

$x\left(t\right)=2t-1$

$y\left(t\right)=t^5-2$