Lesson: Equation of a Hyperbola

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

Sample Question Videos

  • 02:18
  • 02:27

Worksheet: Equation of a Hyperbola • 10 Questions • 2 Videos

Q1:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a garden. The hedge will follow the asymptotes 𝑦 = 3 4 𝑥 and 𝑦 = − 3 4 𝑥 , and its closest distance to the centre fountain is 20 yards. Find the equation of the hyperbola.

Q2:

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 𝑦 = 2 3 𝑥 and 𝑦 = − 2 3 𝑥 , and its closest distance to the center fountain is 12 yards. Find the equation of the hyperbola.

Q3:

Suppose that we model an object’s trajectory in the solar system by a hyperbolic path in the coordinate plane. The 𝑥 -axis is a line of symmetry of this hyperbola. The object enters in the direction of 𝑦 = 0 . 5 𝑥 + 2 and leaves in the direction 𝑦 = − 0 . 5 𝑥 − 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 asymptote’s equations, find the equation of the object’s path.

Q4:

Suppose we model an asteroid’s trajectory by a hyperbolic path in the coordinate plane. The 𝑥 -axis is a line of symmetry of this hyperbola, and the object enters in the direction of 𝑦 = 1 3 𝑥 − 1 and leaves in the direction 𝑦 = − 1 3 𝑥 + 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 object’s path.

Q5:

Suppose we model an asteroid’s trajectory by a hyperbolic path in the coordinate plane. The 𝑥 -axis is a line of symmetry of this hyperbola, and the object enters in the direction of 𝑦 = 𝑥 − 2 and leaves in the direction 𝑦 = − 𝑥 + 2 . 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 object’s path.

Q6:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a garden. The hedge will follow the asymptotes 𝑦 = 2 𝑥 and 𝑦 = − 2 𝑥 , and its closest distance to the centre fountain is 6 yards. Find the equation of the hyperbola.

Q7:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a garden. The hedge will follow the asymptotes 𝑦 = 𝑥 and 𝑦 = − 𝑥 , and its closest distance to the centre fountain is 5 yards. Find the equation of the hyperbola.

Q8:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a yard. The hedge will follow the asymptotes 𝑦 = 1 2 𝑥 and 𝑦 = − 1 2 𝑥 , and its closest distance to the centre fountain is 10 yards. Find the equation of the hyperbola.

Q9:

Write the equation of the rectangular hyperbola passing through ( 1 , 1 ) with asymptotes meeting at ( 3 , − 4 ) .

Q10:

Suppose that we model an object’s trajectory in the solar system by a hyperbolic path in the coordinate plane, with its origin at the sun and its units in astronomical units (AU). The 𝑥 -axis is a line of symmetry of this hyperbola. The object enters in the direction of 𝑦 = 3 𝑥 − 9 , leaves in the direction of 𝑦 = − 3 𝑥 + 9 , and passes within 1 AU of the sun at its closest point. Using the equations of the asymptotes, find the equation of the object’s path.
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