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Hyperbolic Mirrors Hyperbolas have interesting reflective properties that make them useful for lenses and mirrors. For example, if a ray of light strikes a convex hyperbolic mirror on a line that would (theoretically) pass through its rear focus, it is reflected through the front focus. This property, and that of the parabola, were used to develop the Cassegrain telescope in \(1672 .\) The focus of the parabolic mirror and the rear focus of the hyperbolic mirror are the same point. The rays are collected by the parabolic mirror, then reflected toward the (common) focus, and thus are reflected by the hyperbolic mirror through the opening to its front focus, where the eyepiece is located. If the equation of the hyperbola is \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\) and the focal length (distance from the vertex to the focus) of the parabola is \(6,\) find the equation of the parabola.

Step by step solution

01

- Identify the properties of the given hyperbola

The equation of the hyperbola is \(\frac{y^2}{9} - \frac{x^2}{16} = 1\). This is a vertical hyperbola with semimajor axis length \(a = \sqrt{9} = 3\) and semiminor axis length \(b = \sqrt{16} = 4\). The distance from the center to each focus is given by \(c = \sqrt{a^2 + b^2} = \sqrt{9 + 16} = \sqrt{25} = 5\). Therefore, the foci are located at \((0, \pm 5)\).

02

- Understand the parabolic mirror's focal length

The parabolic mirror shares the same focus point as the hyperbolic mirror, which is at \((0, 5)\). The focal length (distance from the vertex to the focus) given for the parabola is \6\ units.

03

- Determine the vertex form of the parabolic equation

A parabola with its vertex at the origin \((0, 0)\) and opening upwards can be expressed as \(y = \frac{1}{4f}x^2\), where \(f\) is the focal length. In this case, \(f = 6\).

04

- Substitute the focal length into the parabolic equation

Using the focal length, the equation becomes \(y = \frac{1}{4 \times 6} x^2 = \frac{1}{24}x^2\). Therefore, the equation of the parabola is \(y = \frac{1}{24}x^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

hyperbola

A hyperbola is a type of conic section formed by the intersection of a plane with a double cone. The defining feature of a hyperbola is that it consists of two separate curves, called branches, that mirror each other. The standard form of a hyperbola's equation is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for hyperbolas that open vertically, as in our exercise example, or \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for those that open horizontally. Each branch approaches a pair of straight lines called asymptotes but never actually meets them. Important properties include: