What Is the Focal Length of a Hyperbola?

TL;DR

The focal length of a hyperbola is equal to the square root of the sum of the squares of its two denominators in the equation, specifically f = √(a² + b²). This relationship holds for hyperbolas that open in both horizontal and vertical orientations.

Transcript

In the last video, I told you that if I had a hyperbola with the equation x squared over a squared minus y squared over b squared is equal to 1, that the focal distance for this hyperbola is just equal to the square root of the sum of these two numbers. The square root of a squared plus b squared. In this video I really just want to show you that. ... Read More

Key Insights

• 🤗 A hyperbola with the equation x^2/a^2 - y^2/b^2 = 1 opens left and right, while a hyperbola with y^2/b^2 - x^2/a^2 = 1 opens upwards and downwards.
• 😥 The difference of distances to the two foci points in a hyperbola is always equal to 2a.
• 🫚 The focal length of a hyperbola is calculated as the square root of the sum of the denominators in its equation.

Questions & Answers

Q: How can the opening direction of a hyperbola be determined?

The opening direction of a hyperbola can be determined by the signs of the x and y terms in its equation. If the x term is positive, the hyperbola opens left and right. If the y term is positive and the x term is negative, the hyperbola opens upwards and downwards.

Q: What is the definition of a hyperbola?

A hyperbola is the locus of all points where the difference of distances to the two foci points is a constant number. This difference, denoted as d1 - d2, is equal to 2a.

Q: How can the focal length of a hyperbola be calculated?

The focal length of a hyperbola is equal to the square root of the sum of the denominators in its equation. This can be derived from the equation f^2 - a^2 = b^2, where f is the focal length.

Q: What is the graphical representation of a hyperbola?

A hyperbola intersects the x-axis at (a, 0) and (-a, 0), and its foci points are located at (f, 0) and (-f, 0). It also has asymptote lines given by y = +/- b/a.

Summary & Key Takeaways

• The video explains that a hyperbola with the equation x^2/a^2 - y^2/b^2 = 1 opens left and right, with asymptotes given by the lines y = +/- b/a.

• The video demonstrates how to graphically represent a hyperbola and locate its foci points.

• The video provides a proof that the difference of distances to the two foci points is equal to 2a, which is used to derive the equation of a hyperbola.