Problem 39 - Precalculus

Question

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (±1,0)\(;\) asymptotes: \(y=\pm 5 x\)

Step-by-Step Solution

Answer

The standard form of the equation of the hyperbola is \(x^2 - y^2/25 = 1\).

Step 1: Identify the value of a from the vertices

The standard form of a hyperbola centered at the origin with the transverse axis along the x-axis is \(x^2/a^2 - y^2/b^2 = 1\). The vertices (±1,0) indicate that \(a = 1\).

Step 2: Determine the value of b/a from the asymptotes

The slope of the asymptotes of the hyperbola \(x^2/a^2 - y^2/b^2 = 1\) is given by ±b/a. In this case, the asymptotes slope given by \(y=\pm 5x\) indicates that \(b/a = 5\). Thus \(b = 5*a = 5*1 = 5\).

Step 3: Write the equation of the hyperbola

The standard form of the hyperbola equation is \(x^2/a^2 - y^2/b^2 = 1\). Replacing a = 1 and b = 5 in our equation we get \(x^2/1^2 - y^2/5^2 = 1\), which simplifies to \(x^2 - y^2/25 = 1\).

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola provides a structured equation that describes the set of all points that form the hyperbola in a plane. This equation differs from that of circles and ellipses because it involves subtraction, not addition. For a hyperbola centered at the origin, the standard form can be written as: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] when the transverse axis is horizontal, or: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] when the transverse axis is vertical. The constant 'a' represents the distance from the center to each vertex along the transverse axis, and 'b' is related to the distance affecting the shape and steepness of the hyperbola. In this specific exercise, the transpose axis is horizontal, as we can tell from the vertices and their coordinates. By correctly placing these values, we accurately outline the pathway of the hyperbola on a graph.

Asymptotes of a Hyperbola

Asymptotes play a crucial role in defining the behavior and shape of hyperbolas. They are the lines that the hyperbola approaches but never actually intersects, giving the hyperbola its definitive shape. In mathematical terms, these asymptotes act as boundary lines that show where the hyperbola "opens up" on the graph.
For a hyperbola with a horizontal transverse axis, the equations of the asymptotes are given by: \[ y = \pm \frac{b}{a} x \] where \( \frac{b}{a} \) represents the slope of the asymptotes. In this scenario, given the asymptotes \( y = \pm 5x \), we understand that \( b = 5a \). This information helps in determining the value of 'b' in the standard form. Hyperbolas tend to align more closely with these asymptotes as they stretch out toward infinity, giving them their characteristic open curves.

Vertices of a Hyperbola

Vertices are specific points on the hyperbola that directly relate to its overall size and shape. They are the points at which the hyperbola is closest to its center. The position of the vertices determines the 'a' value in the standard equation of a hyperbola. If the hyperbola is centered at the origin with vertices located at (±a,0) for a horizontal transverse axis, then these points provide the exact length from the center to the vertices along this axis.
In the given problem, the vertices are located at (±1,0), which tells us that 'a' is 1. These vertices are essential in forming the standard equation of the hyperbola and help us graph its general direction. Accurate location of these points is crucial for understanding the true scale and orientation of the hyperbola's path in a plane.