AP Precalculus / Unit 4: Functions Involving Parameters, Vectors, and Matrices
4.4 Parametrically Defined Circles and Lines
Notes
- Circle of radius $r$ centered at $(h,k)$: $x=h+r\cos t$, $y=k+r\sin t$, $0\le t\le2\pi$ (counterclockwise).
- Line segment from $(x_1,y_1)$ to $(x_2,y_2)$: $x=x_1+(x_2-x_1)t$, $y=y_1+(y_2-y_1)t$, $0\le t\le1$.
Practice
$x=3\cos t$, $y=3\sin t$ traces:
$x^2+y^2=9$.
The segment from $(0,0)$ to $(4,2)$ at $\displaystyle t=\frac{1}{2}$ is at:
Halfway.
$x=1+2\cos t$, $y=-3+2\sin t$ is a circle centered at:
$(h,k)=(1,-3)$.
Ready for more? Take the Unit 4 test.