Solve r=2a(2-cosθ)

$$r = 2 a ( 2 - \cos \theta )$$

Solve for a

$$a=-\frac{r}{2\left(\cos(\theta )-2\right)}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$2a\left(2-\cos(\theta )\right)=r$$

Use the distributive property to multiply 2a by 2-\cos(\theta ).

$$4a-2\cos(\theta )a=r$$

Combine all terms containing a.

$$\left(4-2\cos(\theta )\right)a=r$$

The equation is in standard form.

$$\left(-2\cos(\theta )+4\right)a=r$$

Divide both sides by 4-2\cos(\theta ).

$$\frac{\left(-2\cos(\theta )+4\right)a}{-2\cos(\theta )+4}=\frac{r}{-2\cos(\theta )+4}$$

Dividing by 4-2\cos(\theta ) undoes the multiplication by 4-2\cos(\theta ).

$$a=\frac{r}{-2\cos(\theta )+4}$$

Divide r by 4-2\cos(\theta ).

$$a=\frac{r}{2\left(-\cos(\theta )+2\right)}$$

Solve for r

$$r=2a\left(-\cos(\theta )+2\right)$$