Step 1
Consider the given:
eccentricity, \(e = 2\)
directrix, \(x = 4\)
If the eccentricity is greater than 1 then we get a hyperbola.
And if the directrix is the line \(x = d,\ \text{then we have,}\ r=\ \frac{ed}{1\ +\ e\ \cos\ \theta}\)
Step 2
According to question:
Substitute the value of eccentricity and directrix in polar equation of hyperbola.
\(r=\ \frac{2(4)}{1\ +\ 2\ \cos\ \theta}\)

\(r=\ \frac{8}{1\ +\ 2\ \cos\ \theta}\)

\(r=\ \frac{8}{1\ +\ 2\ \cos\ \theta}\)