Lesson: Equation of a Hyperbola

A gardener wants to grow a hedge around a fountain in their yard. They decide to plant the hedge along a hyperbola, where one of the foci of the hyperbola is at the fountain. At its closest, the hedge will be a distance of 20 yards from the fountain. Using a coordinate system whose origin is at the fountain and with units in yards, the path of the hedge has asymptotes and . Find the equation of the hyperbola in this coordinate system.

A hedge is to be constructed in the shape of a hyperbola near a fountain at the center of a park. The hedge will follow the asymptotes and , and its closest distance to the center fountain is 12 yards. Find the equation of the hyperbola.

Suppose that we model an object’s trajectory in the solar system by a hyperbolic path in the coordinate plane. The -axis is a line of symmetry of this hyperbola. The object enters in the direction of and leaves in the direction . The sun is positioned at the origin and the object passes within 1 AU (astronomical unit) of the sun at its closest. Using the asymptote’s equations, find the equation of the object’s path.