Animated Solution for Mathematics - Conic Sections: The eccentricity of an ellipse, with its centre at the origin, is $1/2$. If one of the directrices is $x=4$, then the equation of the ellipse is:

When you look at an ellipse, do not just see an oval. See a locus of points defined by a beautiful, rigid dance between a focus and a directrix. Our problem provides a center at the origin, an eccentricity of $e=\frac{1}{2}$, and a directrix at $x=4$.

Because the directrix $x=4$ is a vertical line, it forces our ellipse to stretch horizontally. The directrix is always perpendicular to the major axis; therefore, the major axis must lie perfectly along the $x$-axis.

This confirms the standard form of our equation:

For a horizontal ellipse, the directrix is defined by the equation $x=\frac{a}{e}$. This relationship dictates that the distance from the center to the directrix is exactly the semi-major axis divided by the eccentricity.

Given $x=4$ and $e=\frac{1}{2}$, we substitute these values into the formula:

Solving for $a$, we multiply $4$ by $\frac{1}{2}$, yielding $a=2$. Squaring this result gives us the value for the denominator:

We utilize the fundamental relationship that binds $a$, $b$, and $e$ together:

Substituting our known values into this equation:

First, square the eccentricity to get $\frac{1}{4}$, then subtract this from $1$ to obtain $\frac{3}{4}$. Calculating the final value for $b^2$:

We have determined our parameters to be $a^2=4$ and $b^2=3$. Substituting these into our standard form, we obtain:

To match the standard linear form, we clear the denominators by multiplying the entire equation by the least common multiple, which is $12$:

Distributing the $12$ across the terms, we arrive at the final equation: $3x^2+4y^2=12$