The eccentricity of an ellipse, with its centre at the origin, is $$\frac{1}{2}$$. If one of the directrices is $$x = 4$$, then the equation of the ellipse is

Let the standard equation of the ellipse centered at the origin be:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

The eccentricity of the ellipse is given as:

$$ e = \frac{1}{2} $$

The equation of one of its directrices is given as:

$$ x = 4 $$

For a standard ellipse, the equation of the directrix parallel to the y-axis is given by the formula:

$$ x = \frac{a}{e} $$

Equate the given directrix value to the formula:

$$ \frac{a}{e} = 4 $$

$$ \frac{a}{\left(\frac{1}{2}\right)} = 4 $$

$$ a = 2 $$

$$ a^2 = 4 $$

For an ellipse, the relationship between the semi-major axis, semi-minor axis, and eccentricity is given by:

$$ b^2 = a^2(1 - e^2) $$

$$ b^2 = 4\left(1 - \left(\frac{1}{2}\right)^2\right) $$

$$ b^2 = 4\left(1 - \frac{1}{4}\right) $$

$$ b^2 = 3 $$

$$ \frac{x^2}{4} + \frac{y^2}{3} = 1 $$

$$ 3x^2 + 4y^2 = 12 $$