# Parametrization of Witch of Agnesi

1. Feb 15, 2015

### PWiz

1. The problem statement, all variables and given/known data

The question is completely described in the photo.
2. Relevant equations
Trigonometric translation properties

3. The attempt at a solution
The problem is in two dimensions, so I'm ignoring the z coordinates. For a circle centered at (0,a), the position vector of P is $(a$ $sin(θ),a-a$ $cos(θ))$ (by taking into consideration what theta is in this problem) since $a$ $cos(\frac{3π}{2}+θ)=a$ $sin(θ)$ and $a+a$ $sin(\frac{3π}{2}+θ)=a-a$ $cos(θ)$ .
Therefore, the y coordinates of R should be $a-a$ $cos(θ)$ . R's x coordinate equals to the x coordinate of Q, which is given by $x=\frac{2a}{m}$ , where $m=\frac{1-cos(θ)}{sin(θ)}$ .
So my answer is $γ(θ) = (\frac{2asin(θ)}{1-cos(θ)},a-cos(θ))$. However, the correct answer is $γ(θ) = (2a$ $cot(θ), a(1-cos(2θ))$. Where did I go wrong?

2. Feb 15, 2015

### Orodruin

Staff Emeritus
What you are calling $\theta$, your solution manual calls $2\theta$. Make this substitution and use some trigonometric identities and you will get the same. This is of course only a different parametrisation. You are also missing an a multiplying the cos in your y coordinate.

3. Feb 15, 2015

### PWiz

Oops, I made a typo at the end there. And yes, if it is as you say, then my answer would be correct, although why my book refers to $\theta$ as $2\theta$ is beyond me. Thank you.