How do I find the directrix of a hyperbola?

To find the directrix of a hyperbola, you need to know the equation of the hyperbola and its eccentricity. The directrix is a line that is perpendicular to the transverse axis and located a certain distance away from the center of the hyperbola.

Here are the steps to find the directrix of a hyperbola:

Determine the equation of the hyperbola in standard form:
(x-h)²/a² - (y-k)²/b² = 1 or (y-k)²/a² - (x-h)²/b² = 1

The center of the hyperbola is represented by the point (h, k).

Find the value of "a" and "b":

"a" represents the distance from the center to the vertices along the transverse axis.
"b" represents the distance from the center to the vertices along the conjugate axis.
Calculate the eccentricity (e):

The eccentricity of a hyperbola is given by the formula: e = √(a² + b²) / a
This value will help determine the position and equation of the directrix.
Determine the position of the directrix:

If the hyperbola is horizontally aligned (x-h)²/a² - (y-k)²/b² = 1:

The directrix is a horizontal line given by the equation: y = k ± (a / e)
The sign is determined by the direction of the opening of the hyperbola.
If the hyperbola opens to the left or right, use the plus sign (+) for the directrix equation.
If the hyperbola opens upwards or downwards, use the minus sign (-) for the directrix equation.
If the hyperbola is vertically aligned (y-k)²/a² - (x-h)²/b² = 1:

The directrix is a vertical line given by the equation: x = h ± (a / e)
The sign is determined by the direction of the opening of the hyperbola.
If the hyperbola opens upwards or downwards, use the plus sign (+) for the directrix equation.
If the hyperbola opens to the left or right, use the minus sign (-) for the directrix equation.
By following these steps, you can determine the equation and position of the directrix for a hyperbola.