The latus rectum of an ellipse is the chord through either...

The latus rectum of an ellipse is the chord through either Focus perpendicular to the major axis. Show that the length of the latus recta of ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1$ is given by the formula $\left(2 b^{2}\right) / a$.

Key Concepts

Latus Rectum

The latus rectum of an ellipse is a chord that passes through one of the foci and is perpendicular to the major axis. It provides a measure of the ellipse's width at the focus and is instrumental in various geometric and analytic calculations. Its length, often derived using the ellipse’s parameters, reflects the interplay between the major and minor axes, and in this specific context is given by the formula (2b²)/a.

Major Axis

The major axis of an ellipse is the longest diameter that passes through its center and both foci. It defines the direction in which the ellipse is extended the most, and its endpoints are referred to as vertices. The length of the major axis is 2a, which is critical in understanding the ellipse’s dimensions and other related properties.

Focus

In the context of an ellipse, the foci are two fixed points located along the major axis such that the sum of the distances from any point on the ellipse to these foci remains constant. The distance from the center to a focus is given by c, where c² = a² - b², a relation that is fundamental to the ellipse's geometric properties.

Ellipse

An ellipse is a conic section defined as the set of all points in a plane for which the sum of the distances to two fixed points (the foci) is constant. Its standard equation, x²/a² + y²/b² = 1, represents an ellipse oriented along the coordinate axes, where 'a' and 'b' determine the lengths of the semi-major and semi-minor axes, respectively.

Please give Ace some feedback