# Thread: Parametric equation for a curve

1. ## Parametric equation for a curve

If a = 6 and b = 5, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle θ as the parameter. The line segment AB is tangent to the larger circle.
x(θ) = ?
y(θ) = ?
and θ is in between 0 and 2pi

i found the picture on the internet, this isn't my assignment but the curve on my assignment is the exact same as #4 from here:
http://www.iit.edu/~maslanka/Rec3F08.pdf

i tried doing this:
x = a + b
y = asin(θ)

but i didn't knwo what to do from there. add them maybe? but how would you get rid of sin(θ)? i'm confused lol. please help someone.

2. Hello, CenturionMonkey!

If $a = 6$ and $b = 5$, find parametric equations for the curve that consists
of all possible positions of the point $P$ in the figure, using $\theta$
as the parameter. .The line segment $AB$ is tangent to the larger circle.

. . $\begin{array}{ccc}x&=& f(\theta) \\ y &=& g(\theta) \end{array}\quad \text{for }0 \leq\theta \leq 2\pi$

I won't try to copy the diagram this time.

In right triangle $OAB\!:\;\;\sec\theta \:=\:\frac{OB}{OA} \:=\:\frac{x}{6} \quad\Rightarrow\quad x \:=\:6\sec\theta$

In the smallest right triangle: . $\sin\theta \:=\:\frac{y}{5} \quad\Rightarrow\quad y \:=\:5\sin\theta$

Therefore: . $\begin{Bmatrix}x &=& 6\sec\theta \\ y &=& 5\sin\theta \end{Bmatrix}$

3. oh yeah that makes perfect sense. I didn't realize that was a right triangle. lol silly me. thanks