90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

Verified step by step guidance

1. Recall the standard form of a hyperbola centered at the origin with the transverse axis along the x-axis: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
2. Identify the coordinates of the foci of the hyperbola, which are located at \((\pm c, 0)\), where \(c = a e\) and \(e\) is the eccentricity defined by \(e = \frac{c}{a}\).
3. Write the equation of the latus rectum, which is the focal chord perpendicular to the major axis. Since the major axis is along the x-axis, the latus rectum is a vertical line through the focus at \(x = c\).
4. Substitute \(x = c\) into the hyperbola equation to find the \(y\)-coordinates of the points where the latus rectum intersects the hyperbola: \(\frac{c^2}{a^2} - \frac{y^2}{b^2} = 1\).
5. Solve for \(y^2\) to get \(y^2 = b^2 \left( \frac{c^2}{a^2} - 1 \right)\), then use the relationship \(c^2 = a^2 + b^2\) to simplify and find the length of the latus rectum as \$2|y| = \frac{2b^2}{a} = 2b \sqrt{1 - e^2}$.

Key Concepts

Definition of a Hyperbola and Its Parameters

A hyperbola is a conic section defined as the set of points where the difference of distances to two foci is constant. It is characterized by parameters a (distance from center to vertices), b (related to the conjugate axis), and eccentricity e, which measures the deviation from circularity. Understanding these parameters is essential for analyzing focal chords and latus rectum.

Focal Chords and the Latus Rectum

A focal chord is any chord passing through a focus of the conic. The latus rectum is a special focal chord perpendicular to the major axis, intersecting the curve at two points. Its length relates directly to the conic’s parameters and is crucial for proving properties involving distances and eccentricity.

Relationship Between Eccentricity, a, and b in a Hyperbola

For a hyperbola, the eccentricity e satisfies the relation e² = 1 + (b²/a²). This connects the transverse axis length a, conjugate axis length b, and eccentricity e. Using this relationship allows rewriting expressions involving b²/a in terms of e, which is key to proving the formula for the latus rectum length.