Problem 45 - Precalculus: Mathematics for Calculus

Question

Find an equation for the hyperbola that satisfies the given conditions. Asymptotes: \(y=\pm x,\) hyperbola passes through \((5,3)\)

Step-by-Step Solution

Answer

Equation: \(x^2 - y^2 = 16\).

Step 1: Identify the Hyperbola Type

The asymptotes given are \(y = \pm x\). Such asymptotes indicate a hyperbola centered at the origin (0,0) and oriented diagonally, with axes rotated 45 degrees. However, for simplicity without rotation, these could also suggest the use of a standard hyperbola centered at the origin if we match slopes where \(a = b\).

Step 2: Write the Standard Form of the Hyperbola

Given the asymptotes \(y = \pm x\), the form of the hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\). This simplifies to \( x^2 - y^2 = a^2\) by multiplying through by \( a^2 \). Here, \(a = b\) due to matching slopes of the asymptotes.

Step 3: Substitute the Point into the Equation

The hyperbola passes through the point (5, 3). Substitute \((x, y) = (5, 3)\) into \(x^2 - y^2 = a^2\): \[ 5^2 - 3^2 = a^2 \] Calculate: \[ 25 - 9 = a^2 \] \[ a^2 = 16 \].

Step 4: Final Hyperbola Equation

With \(a^2 = 16\), the equation of the hyperbola is:\[ x^2 - y^2 = 16 \].

Key Concepts

Asymptotes

Asymptotes are crucial for understanding the behavior of hyperbolas. They are straight lines that the branches of a hyperbola approach but never actually intersect. In essence, asymptotes guide the shape and direction of the hyperbola.
For a hyperbola centered at the origin, the asymptotes are determined by the equation \( y = rac{b}{a}x \) and \( y = -\frac{b}{a}x \). In our specific case, the asymptotes given are \( y = \pm x \). This tells us that \( a = b \), since the slopes of the asymptotes are \( \pm 1 \) — implying matching horizontal and vertical stretches.

• They help determine the equation of the hyperbola.
• Provide the slopes of the intersecting lines that form the hyperbola's boundaries.
• In this example, the slopes are \(+1\) and \(-1\), indicating equal stretching.

Understanding asymptotes is integral for finding the hyperbola's equation, as they directly relate to the axes of the hyperbola.

Standard Form of Hyperbola

The standard form of a hyperbola equation is essential for organizing and solving hyperbolic problems. It's a formula where you can see how values for \(x\) and \(y\) relate to one another within the hyperbola's context.
In general, for a hyperbola centered at the origin, the standard forms are:

• Horizontal hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Vertical hyperbola: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

In our scenario, since \( a = b \), the equation simplifies due to the symmetry dictated by the asymptotes. The asymptotic condition \( y = \pm x \) suggests a simplified form, leading directly to the expression: \( x^2 - y^2 = a^2 \).
This equation gives us a clear path to find the specific hyperbola by helping us calculate the \(a\) value using given points on the hyperbola.

Hyperbola Centered at Origin

When discussing a hyperbola centered at the origin, we're looking at a simple yet foundational type of hyperbola. Centering at the origin, (0,0), means that the hyperbola is symmetric about both the x-axis and y-axis.
This symmetry is reflected in the simplicity of the standard form equations. When a hyperbola is mathematically centered at the origin, its properties allow us to easily describe and solve related problems:

• Consistent focal points equidistant from the center.
• Symmetrical behavior across both x and y directions.
• Easy to apply both algebraic and geometric principles due to central positioning.

The problem given highlights a case where the hyperbola, due to these central properties, allows straightforward solving, especially when plugging in points to find specific parameters like \(a^2\). This specific exercise showcases how centering eliminates more complicated translations or rotations, making the equations more manageable.