Q. 11. - Calculus

Question

When A > B, the directrices of the ellipse with equation $\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$are the lines with equations $x=\pm \frac{A}{e}$, where e is the eccentricity of the ellipse. What are the equations for the directrices when A < B?

Step-by-Step Solution

Answer

The required answer is $y=\frac{B}{e},y=-\frac{B}{e}$where $e$is the eccentricity of the ellipse.

Step 1: Given information

The given ellipse is $\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$

Step 2: The objective is to write the directrices of the ellipse.

Ellipse is a plane curve with the sum of its periphery distances from two fixed points, the foci, being equal. An ellipse is a subset of a circle.

Every ellipse has two directrices.

The equation is $\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$

The directrices of the ellipse $\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$when $A<B$are the horizontal lines.

They are $y=\frac{B}{e}$and $y=-\frac{B}{e}$where $e$is the eccentricity of the ellipse.