Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.4.92

Chapter 12, Problem 12.4.92

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 the parabola y ² =4px or x ² =4py is 4|p|.

Verified step by step guidance

Recall the standard form of the parabola and its focus. For the parabola \(y^{2} = 4px\), the focus is at the point \((p, 0)\).

Understand that the latus rectum is the focal chord perpendicular to the axis of symmetry. Since the parabola opens along the x-axis, the latus rectum is a vertical line passing through the focus at \(x = p\).

Find the points where the line \(x = p\) intersects the parabola by substituting \(x = p\) into the parabola equation: \(y^{2} = 4p(p) = 4p^{2}\).

Solve for \(y\) to get the two intersection points: \(y = 2p\) and \(y = -2p\). These points are \((p, 2p)\) and \((p, -2p)\).

Calculate the length of the latus rectum as the distance between these two points, which is the difference in their \(y\)-coordinates: \(|2p - (-2p)| = 4|p|\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parabola and its Standard Form

A parabola is a conic section defined as the set of points equidistant from a fixed point (focus) and a fixed line (directrix). The standard forms y² = 4px or x² = 4py represent parabolas opening right/left or up/down, where p is the distance from the vertex to the focus.

Focal Chord and Latus Rectum

A focal chord is any chord passing through the focus of the parabola. The latus rectum is a special focal chord perpendicular to the axis of symmetry (major axis) of the parabola. It connects two points on the curve through the focus and is key to understanding parabola geometry.

Length of the Latus Rectum

The length of the latus rectum is the distance between the two points where the focal chord perpendicular to the axis intersects the parabola. For y² = 4px or x² = 4py, this length is always 4|p|, derived by substituting the focus coordinates into the parabola equation and calculating the chord length.

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Textbook Question

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Textbook Question

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