Deriving the polar equation of a parabola with focus at the origin and directrix at $y=-p$

Question

How is the following result of a parabola with focus $F(0,0)$ and directrix $y=-p$, for $p \gt 0$ reached? It is said to be $$r(\theta)=\frac{p}{1-\sin \theta} $$

I started by saying the the standard equation of a parabola, in Cartesian form is $y= \frac{x^2}{4p} $, where $p \gt 0 $ and the focus is at $F(0,p)$ and the directrix is $y=-p$. So for the question above, would the equation in Cartesian form be $$y= \frac{x^2}{4 \cdot \left(\frac{1}{2}p\right)}=\frac{x^2}{2p}?$$

I thought this because the vertex is halfway between the directrix and the focus of a parabola.

Then I tried to use the facts: $$r^2 = x^2 +y^2 \\ x =r\cos\theta \\ y=r\sin\theta.$$

But I couldn't get the form required, any corrections, or hints?

Cheers.

Answer 1

No, that can't work, because you've shown that these parabolae have different foci.

Since there are lots of parabolae ($x^2=4ay$, $y^2=4ax$, any rotation and translation of one of these), it's easier to work backwards. So let's do that: $\sin{\theta} = y/r$, so $$ p= r(1-\sin{\theta}) = r-y, $$ so $$ x^2+y^2 = r^2 = (p+y)^2 = p^2+2py+y^2, $$ and $$ x^2 = 2py+p^2. $$ In particular, if we look more closely, we notice that $$ \sqrt{x^2+y^2} = p+y $$ is the definition of a parabola: the distance from $(0,0)$ to the point is the same as $y+p$, the distance from the horizontal line $y=-p$.

Answer 2

The equation of the parabola you want is

$$ y = \frac{x^2}{2p} - \frac{p}{2} $$

Substituting

$$ x = r \cos \theta $$ $$ y = r \sin \theta $$

gives us

$$ \frac{\cos^2\theta}{2p} r^2 - (\sin \theta) r - \frac{p}{2} = 0 $$

If you solve this quadratic expression for $r$, and use the identity $\sin^2\theta + \cos^2\theta = 1$ twice, you should obtain the expression you want.

Answer 3

The equation for a parabola in the complex plane is

$$z=\frac{1}{2}p(u+i)^2\\ y=pu\\ x=\frac{1}{2}p(u^2-1) $$

I think you would have to say

$$r=|z|\\ \theta=\arg(z)$$

to get the true polar form.

Ref: Zwikker, C. (1968). The Advanced Geometry of Plane Curves and Their Applications, Dover Press.

Answer 4

For a parabola with $F(0,0)$ and directrix $y=-p$, first write a “distance equation” relating any point on the parabola, $P(x,y)$, to $F$ and to the directrix, $D$.

A parabola can be defined by its locus: $distanceFP = distanceDP$.

$distanceFP = \sqrt{(x-0)^2+(y-0)^2} = \sqrt{x^2+y^2}$

$distanceDP = y-(-p) = y+p$

So, for $P(x,y)$ on the parabola with $F(0,0)$ and directrix $y=-p$, we can write $$\sqrt{x^2+y^2}=y+p …(1)$$ For polar cords, we know that $x^2+y^2=r^2$, and $y=r sin\theta$

Therefore (1) becomes $$\sqrt{r^2}=r sin\theta + p$$ $$r = r sin\theta + p$$ $$r – r sin\theta = p$$ $$r(1 – sin\theta) = p$$ Hence $$r = \frac{p}{1 – sin\theta}$$