Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0,±3)\(;\) asymptotes: \(y=\pm 3 x\)

Step-by-Step Solution

Answer

The standard form of the equation of the hyperbola is \(\frac{y^2}{9} - \frac{x^2}{81} = 1\).

Step 1: Identifying a and b

A hyperbola that opens left and right has a standard form of \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), while one that opens up and down has a standard form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Since the vertices of the hyperbola are placed vertically at (0,±3), we know that our hyperbola opens up and down. 'a' is the distance from the center to each vertex. Given that our vertices are (0,±3), the value of 'a' is 3.

Step 2: Find b from the asymptotes

The equation of the asymptotes of a hyperbola has the form y = ±(b/a)x. In this case, the asymptotes are given as \(y=\pm 3x\), which indicates b/a = 3. We already have a = 3 from Step 1, so we solve this equation for b: \(3 = \frac{b}{3}\) which gives b = 9.

Step 3: Write down the equation of the hyperbola

Now, we can write down equation for the hyperbola using the identified values of a and b. Plugging a = 3 and b = 9 into the standard form of the equation of the hyperbola gives \(\frac{y^2}{9} - \frac{x^2}{81} = 1\).