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What is the parametric form of the curve above? If I had to solve it, what I would say is that the first step is to complete the square. However, where would I go from there?

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You have a good first step --- keep going. How would you parametrize a circle of radius $r$ and center $(h,k)$? – John Engbers May 29 '12 at 20:16
Not exactly sure. But i know for this question, there is no z-parameter, and therefore it isnt a sphere, but probably a cylinder? Not sure how that helps me though.. – Nick May 29 '12 at 20:18
In 2D it is a circle; in 3D a cylinder. How do you paramatrize a circle of radius $r$ centered at $(0,0)$? – John Engbers May 29 '12 at 20:19
x=0+t and y=0+t? – Nick May 29 '12 at 20:20
There are various parametrizations. The mot popular one involves trig functions. The parametric equations $x=t$, $y=t$ describe the line $x=y$. – André Nicolas May 29 '12 at 20:22
up vote -1 down vote accepted

You can give a rational parametrization using stereographical projection, as follows:

The point $(0,0)$ is on the curve. Since the equation is quadratic, for each slope $t$, the line $y=tx$ must intersect the curve at a second point. If you plug $y=tx$ into the equation (and you assume $x\neq 0$), then you should get that the coordinates of the second point of intersection are $$x=\frac{2(2t-1)}{1+t^2},\quad y=\frac{2t(2t-1)}{1+t^2}.$$ Are those all the points on the curve? No, but almost all. If $(x_0,y_0)\neq (0,0)$ is on the curve, then consider the line $L$ from $(0,0)$ to $(x_0,y_0)$:

  • If the line $L$ is vertical, then $x_0=0$, and from the equation we have $$y_0^2-4y_0=0$$ Thus, $y_0=0$ (not allowed, by assumption), or $y_0=4$. Hence $(x_0,y_0)=(0,4)$.

  • Otherwise, the line $L$ is not vertical, and $L$ is of the form $y=tx$ and $(x_0,y_0)$ is as given above: $$(x_0,y_0)=\left(\frac{2(2t-1)}{1+t^2},\ \frac{2t(2t-1)}{1+t^2}\right).$$

Hence, the points on your curve are $$\{(0,4)\}\cup \left\{\left(\frac{2(2t-1)}{1+t^2},\ \frac{2t(2t-1)}{1+t^2}\right): t\in \mathbb{R}\right\},$$ and that's the parametrization we were looking for (notice $(0,0)$ is on the right hand side, with $t=1/2$).

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Why the -1? Any suggestions for improvement? – Álvaro Lozano-Robledo Jun 27 '12 at 13:27

$$x^2 + y^2 + 2x - 4y = 0$$

$$x^2 + 2x + y^2- 4y = 0$$

$$x^2 + 2x + 1 -1 +y^2- 4y +4 -4= 0$$

$$(x+1)^2 -1 + (y-2)^2-4= 0$$

$$(x+1)^2 + (y-2)^2= 5$$

For a circle with centre at (h,k) and radius r, the a parametric form is $x = h + r\cos(\theta)$ and $y = k + r\sin(\theta)$.

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For those of us who love rational parametrizations, may I suggest "a parametric form" instead of "the parametric form". – André Nicolas May 29 '12 at 20:31
@AndréNicolas Edited. Good Point. – Inquest May 29 '12 at 20:35

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