Polar Conics Example 1 - Sketch r = 3/(2+2cosθ)

In this video, we'll identify the features that will allow us to sketch the equation r = 3/(2+2cosθ).
By comparing the equation

r=\frac{3}{2+2\cos\theta}

with

r=\frac{l}{1+e\cos\theta}

we see that it's not quite in the standard polar form.

However, if we divide both the numerator and the denominator by 2, we get the expression:

r=\frac{\frac{3}{2}}{1+1\cos\theta}

Now from this, we can see that this conic has an eccentricity e = 1, which means it forms a parabola.

From there, we can find the vertex by letting θ = 0, which gives r(0) = ¾.

The semi-latus rectum length l = 1.5, which means at the polar angles of ±π/2, that is directly above and below the focus, we have r = 1.5.

With this data, we can quickly and accurately sketch the curve r = 3/(2+2cosθ)