Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: (0,±8)\(;\) asymptotes: \(y=\pm 4 x\)

Step-by-Step Solution

Answer

The standard form of the hyperbola equation is \(\frac{17x^{2}}{64}-\frac{17y^{2}}{1024}=1\).

Step 1: Find the value of c

The foci of the hyperbola are given as (0,±8). Therefore, the value of c is 8, as c is the distance from the origin to either focus.

Step 2: Determine a and b using asymptotes

The equations of the asymptotes are given as \(y=\pm 4x\). It can be seen that the asymptotes have a slope of ±4, which suggests that the ratio of a to b is 4. Therefore, \(\frac{a}{b}=4\). As we know that the hyperbola is oriented vertically because the foci are vertically aligned, we can say that a=4b.

Step 3: Solve for a and b using the hyperbola equation

We know that for a hyperbola, \(c^{2}=a^{2}+b^{2}\). Using the values of a and b, we can solve for the specific values. Substituting the value of c=8 and \(a=4b\) into the equation, we get \(8^{2} = (4b)^{2} + b^{2}\), which simplifies to \(64 = 16b^{2} + b^{2}\), and further simplifies to \(64=17b^{2}\). Solving for b, we get \(b^{2}=\frac{64}{17}\). Substituting \(b^{2}\) into \(a=4b\) to solve for \(a^{2}\), we get \(a^{2}=16b^{2}=16*\frac{64}{17}=\frac{1024}{17}\).

Step 4: Write the standard form of hyperbola equation

Now that we have the values of \(a^{2}\) and \(b^{2}\), we can write the standard form of the hyperbola equation. Since the hyperbola is oriented vertically (confirmed by the vertical alignment of foci and the slope of asymptotes), the standard form of the hyperbola is \(\frac{x^{2}}{b^{2}}-\frac{y^{2}}{a^{2}}=1\). Substituting those values gives us \(\frac{x^{2}}{\frac{64}{17}}-\frac{y^{2}}{\frac{1024}{17}}=1\). Simplifying the equation gives the final answer: \(\frac{17x^{2}}{64}-\frac{17y^{2}}{1024}=1\).