Consider an object moving in the plane whose location at time t seconds is given by the parametric equations:
x(t)=4cos(πt)
y(t)=3sin(πt).

Assume the distance units in the plane are meters.
(x^2/a^2)+(y^2/b^2)=1
What are a and b?

Question

Consider an object moving in the plane whose location at time t seconds is given by the parametric equations:
x(t)=4cos(πt)
y(t)=3sin(πt). 

Assume the distance units in the plane are meters.
(x^2/a^2)+(y^2/b^2)=1
What are a and b?

parthkohli · Answer

So think of it like this. What would you do with these equations to eliminate the sine and cosine? Here's a hint:$$x = \cos(\pi t)$$$$y = \sin (\pi t)$$$$\Rightarrow x^2 + y^2 = \cos^2 (\pi t) + \sin^2 (\pi t) \tag{square both equations and add}$$But $\cos^2 (\pi t) +\sin^2 (\pi t) = 1$, so$$x^2 + y^2 = 1$$

parthkohli · Answer

It's just the same strategy except here, you've got $x/4 = \cos(\pi t)$ and $y/3 = \sin(\pi t)$.

anonymous · Answer

Right so after solving for y and x I just plug back in to the equation with a and b and solve for those?

mathmale · Answer

Can't solve for y and x.  Rather, follow Parth Kohli's approach, which uses the Pythagorean Theorem to obtain the equivalent of  x^2 + y^2 = 1 and thus eliminates the parameter t.  Show your work, please.

Hint:  What geometric figure is represented by$$\frac{ x^2 }{ a^2}+\frac{ y^2 }{ b^2 }=1?$$

anonymous · Answer

An elipse is shown

parthkohli · Answer

Solve for cosine and sine, actually, then square and add.

anonymous · Answer

I've gotten where I have 1=(x/4+y/3)^2