How to find the directrix of an ellipse?
Understand the Problem
The question is asking how to determine the directrix of an ellipse, which is a line associated with the ellipse that helps define its shape and position. To solve this, one typically uses the standard form of the ellipse equation and the definition of the directrix in relation to the ellipse's foci.
Answer
The directrix for a horizontal ellipse is $x = \pm \frac{a^2}{c}$, and for a vertical ellipse is $y = \pm \frac{b^2}{c}$.
Answer for screen readers
The coordinates of the directrix for a horizontal ellipse are given by $x = \pm \frac{a^2}{c}$, and for a vertical ellipse by $y = \pm \frac{b^2}{c}$, where $c = \sqrt{a^2 - b^2}$.
Steps to Solve
- Identify the standard form of the ellipse
The standard form of an ellipse centered at the origin can be expressed as:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
Here, $2a$ is the length of the major axis, and $2b$ is the length of the minor axis.
- Determine the foci of the ellipse
The foci can be found using the formula:
$$ c = \sqrt{a^2 - b^2} $$
where $c$ is the distance from the center to each focus. The foci are located at points $(\pm c, 0)$ for a horizontally oriented ellipse or $(0, \pm c)$ for a vertically oriented ellipse.
- Write the equation for the directrix
For an ellipse, the directrices can be calculated using the following formula based on the orientation:
- If the ellipse is horizontal, the directrices are located at:
$$ x = \pm \frac{a^2}{c} $$
- If the ellipse is vertical, the directrices are located at:
$$ y = \pm \frac{b^2}{c} $$
- Calculate the directrices
Plug in your values of $a$, $b$, and $c$ into the equations from the previous step to find the coordinates of the directrix.
The coordinates of the directrix for a horizontal ellipse are given by $x = \pm \frac{a^2}{c}$, and for a vertical ellipse by $y = \pm \frac{b^2}{c}$, where $c = \sqrt{a^2 - b^2}$.
More Information
The directrix serves as a line from which the distance to any point on the ellipse is compared to the distance to the foci, maintaining a specific constant ratio. This concept is crucial in the study of conic sections.
Tips
- Confusing the formulas for directrices of ellipses with those of hyperbolas.
- Miscalculating the value of $c$ by incorrectly applying the equation $c = \sqrt{a^2 - b^2}$.
