How to Solve a Parametric Equation with Multiple Parts?

[neshepard]

Homework Statement

So I'm given x(t)=4cos(t) and y(t)=sin(t). In a 4 part question, I 1st need to eliminate the parameter to get the equivalent rectangular equation, 2nd I need to find the slope of the tangent to the curve C-y(x) at t=1, 3rd find the times the tangent to the graph is undefined, and lastly set up the integral to find the distance traveled by the [article during one cycle.

The Attempt at a Solution

1st-Eliminate the parameter:(Hopefully correct?)
(x/4)^2 + y^2 = (sin(t))^2 + (cos(t))^2 = 1
1/4x^2 + y^2 = 1

2nd-What do I do here? Do I need to go to the original parametric equations?

3rd-I know the graph has undefined tangents at (-4,0) and (4,0) but how do I determine the time when these occur?

4th-Integral of distance traveled in one cycle:(Hopefully correct?)
∫√((cos(t))^2 + (-4sin(t))^2) dt from 0 to 2pi.

Can anybody please help me understand since my professor is hauling butt through all this?

 

[LCKurtz]

Homework Statement

So I'm given x(t)=4cos(t) and y(t)=sin(t). In a 4 part question, I 1st need to eliminate the parameter to get the equivalent rectangular equation, 2nd I need to find the slope of the tangent to the curve C-y(x) at t=1, 3rd find the times the tangent to the graph is undefined, and lastly set up the integral to find the distance traveled by the [article during one cycle.

The Attempt at a Solution

1st-Eliminate the parameter:(Hopefully correct?)
(x/4)^2 + y^2 = (sin(t))^2 + (cos(t))^2 = 1
1/4x^2 + y^2 = 1

Good so far.

2nd-What do I do here? Do I need to go to the original parametric equations?

You can if you wish. Do you know the formula for dy/dx in terms of dy/dt and dx/dt?
Or you can find dy/dx by implicit differentiation of your x-y equation

3rd-I know the graph has undefined tangents at (-4,0) and (4,0) but how do I determine the time when these occur?

Look at your parametric equations and see what time gives x = 4 or -4 and y = 0.

4th-Integral of distance traveled in one cycle:(Hopefully correct?)
∫√((cos(t))^2 + (-4sin(t))^2) dt from 0 to 2pi.

Looks OK.

 

[neshepard]

Cool thanks.