In this lesson, we will learn how to analyze and write equations of hyperbolas.

Q1:

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 π¦ = 2 3 π₯ and π¦ = β 2 3 π₯ , and its closest distance to the center fountain is 12 yards. Find the equation of the hyperbola.

Q2:

Suppose that we model an objectβs trajectory in the solar system by a hyperbolic path in the coordinate plane. The π₯ -axis is a line of symmetry of this hyperbola. The object enters in the direction of π¦ = 0 . 5 π₯ + 2 and leaves in the direction π¦ = β 0 . 5 π₯ β 2 . The sun is positioned at the origin and the object passes within 1 AU (astronomical unit) of the sun at its closest. Using the asymptoteβs equations, find the equation of the objectβs path.

Q3:

Suppose we model an asteroidβs trajectory by a hyperbolic path in the coordinate plane. The π₯ -axis is a line of symmetry of this hyperbola, and the object enters in the direction of π¦ = 1 3 π₯ β 1 and leaves in the direction π¦ = β 1 3 π₯ + 1 . The sun is positioned at the origin, and the object passes within 1 AU (astronomical unit) of the sun at its closest such that the sun is one focus of the hyperbola. Give the equation of the objectβs path.

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