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. \(r=\ \frac{2(4)}{1\ +\ 2\ \cos\ \theta}\)

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