Problem 31 - Precalculus with Limits

Question

In Exercises 29-34, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. Vertices: \((\pm1, 0); \quad\) asymptotes: \(y=\pm5x\)

Step-by-Step Solution

Answer

The standard form of the equation of the hyperbola given its characteristics and center at the origin is \[x^{2} - y^{2}/25 = 1\].

Step 1: Using the vertices to find 'a'

The given vertices (\(\pm1, 0\)) tell us that 'a' is equal to the distance from the center of the hyperbola to each vertex – in this case, 1.

Step 2: Using the asymptotes to find 'b'

The equations of the asymptotes are \(y=\pm5x\). This slope is equal to the ratio 'b/a'. Given that 'a' is 1, it follows that 'b' must be equal to the absolute value of the slope, which is 5.

Step 3: Writing the standard form equation of the hyperbola

Using the values for 'a' and 'b' found in steps 1 and 2, it's possible to construct the standard form of the equation for the hyperbola. As the vertices are along the x-axis, the standard form is \[x^{2}/{a^{2}} - y^{2}/{b^{2}} = 1\]. Plugging in the values of 'a' and 'b' gives the equation \[x^{2}/{1^{2}} - y^{2}/{5^{2}} = 1\]. Simplifying, the equation becomes \[x^{2} - y^{2}/25 = 1\].

Key Concepts

Standard Form of a Hyperbola

Hyperbolas are fascinating curves expressed through specific algebraic equations. The standard form of a hyperbola's equation helps us understand its orientation and scale, especially when its center is at the origin (0, 0). There are two main types of hyperbolas:

• Horizontal Hyperbola: The vertices are aligned along the x-axis.
• Vertical Hyperbola: The vertices are aligned along the y-axis.

For a hyperbola centered at the origin, where the vertices are on the x-axis, the standard form of the equation is:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]This equation denotes a horizontal hyperbola. Conversely, if the vertices were aligned along the y-axis, the equation would be:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]In these equations, 'a' and 'b' are separated across different denominators indicating their roles in determining the hyperbola's shape and orientation. The parameters 'a' and 'b' impact the direct distance of the vertices and the slope of the asymptotes.

Vertices of Hyperbola

Vertices are crucial as they mark the closest and farthest points of a hyperbola from its center. For a hyperbola curve, the vertices are linearly aligned with the axes:

• If horizontally aligned: The vertices are at \((\pm a, 0)\).
• If vertically aligned: The vertices are at \((0, \pm a)\).

The distance from the hyperbola's center to each vertex is the value 'a'. This is a characteristic parameter directly derived from the hyperbola's equation. In our example, the vertices are at \((\pm 1, 0)\), confirming their alignment along the x-axis. Therefore, 'a' is equal to 1. Understanding vertices provides insight into the 'openness' of the hyperbola along a given axis.

Asymptotes of Hyperbola

Asymptotes of a hyperbola are diagonal lines that the hyperbola approaches but never intersects. These lines resemble the directions in which the hyperbola expands and are essential for predicting its long-range path.For a hyperbola centered at the origin, asymptotes take the following forms:

• If the hyperbola is horizontal, the asymptotes equations are \(y = \pm \frac{b}{a}x\).
• If it's vertical, the equations are \(y = \pm \frac{a}{b}x\).

In the given problem, the asymptotes are \(y = \pm 5x\), indicating that the ratio \(\frac{b}{a}\) equals 5. With 'a' noted as 1, 'b' must logically be 5. Asymptotes not only convey the slope but also play a significant role in sketching and visualizing the hyperbola’s bounds.