Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form.

Step-by-Step Solution

Answer

The standard form of a hyperbola with a horizontal transverse axis is \(x^2/c^2 - y^2/b^2 = 1\), where 'c' is the distance from the center to a focus and 'b' is the square root of \(c^2 - a^2\), with 'a' being half of the constant difference between distances to foci from any point on the hyperbola.

Step 1: Define the Foci

Let's define the two foci located at \((-c, 0)\) and \((c, 0)\) on the x-axis at equal distances from the origin, the center of the hyperbola. 'c' is the distance of a focus from the center.

Step 2: Applying the Definition

Take a point P on the hyperbola, with coordinates \((x, y)\). According to the definition of a hyperbola, the difference of the distances PF1 and PF2 (where F1 and F2 are the foci) is a constant, which we will denote as \(2a\), with 'a' being half of the difference. It gives us \(PF1 - PF2 = 2a\), where \(PF1 = \sqrt{(x-c)^2 + y^2}\) and \(PF2 = \sqrt{(x+c)^2 + y^2}\).

Step 3: Simplify the Expression

Squaring both sides will remove the square roots. After simplification, it gives us that \(x^2/c^2 - y^2/c^2 - a^2/c^2 = 1\). Suppose \(b^2 = c^2 - a^2\), we have \(x^2/c^2 - y^2/b^2 = 1\), which is the standard equation of a hyperbola with a horizontal transverse axis.