If a directrix of a hyperbola centred at the origin and passing through the point (4, √3) is 5x = 4√5 and its eccentricity is e, then:

Let the equation of the hyperbola be x²/a² - y²/b² = 1.
Since the hyperbola passes through (4, √3), we have 16/a² - 3/b² = 1.
The directrix is given by 5x = 4√5, or x = 4√5/5.
The eccentricity is e.
For a hyperbola, the distance from the center to the directrix is a/e.
Therefore, a/e = 4√5/5.
Also, b² = a²(e² - 1).
Substituting b² in the equation of the hyperbola, we get 16/a² - 3/(a²(e² - 1)) = 1.
16(e² - 1) - 3 = a²(e² - 1).
16e² - 16 - 3 = a²(e² - 1).
16e² - 19 = a²(e² - 1).
Since a/e = 4√5/5, a = 4√5e/5.
Substituting this value of a, we get:
16e² - 19 = (16(5e²)/25)(e² - 1).
16e² - 19 = (16e²/5)(e² - 1).
80e² - 95 = 16e⁴ - 16e².
16e⁴ - 96e² + 95 = 0.
4e⁴ - 24e² + 95/4 = 0.
This equation does not match any of the given options.
However, let's assume the directrix is x = 4√5/5 = a/e.
Then a = 4√5e/5.
The point (4, √3) lies on the hyperbola, so (16/a²) - (3/b²) = 1.
b² = a²(e² - 1).
Substituting, we get:
16/a² - 3/(a²(e² - 1)) = 1.
16(e² - 1) - 3 = a²(e² - 1).
16e² - 19 = a²(e² - 1).
Substituting a = 4√5e/5, we get:
16e² - 19 = ((16 * 5e²)/25)(e² - 1)
80e² - 95 = 16e⁴ - 16e²
16e⁴ - 96e² + 95 = 0.
This equation still does not match any of the given options. There might be an error in the problem statement or the options provided.