Problem 35 - Algebra and Trigonometry

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

The equation of the hyperbola is \(x^2 - y^2 = 16\).

Step 1: Understand the standard form of a hyperbola

The standard form of a hyperbola with asymptotes as lines passing through the origin is given by \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) where its asymptotes are \( y = \pm \frac{b}{a}x \). For \( y = \pm x \) (since \( \pm x = \pm \frac{b}{a}x \)), it follows that \( a = b \).

Step 2: Simplify the equation using the asymptotes condition

Since the asymptotes are \( y = \pm x \), both \( a \) and \( b \) are equal. Hence, \( f(x, y) = \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 \) can be rewritten as \( \frac{x^2 - y^2}{a^2} = 1 \). So, the equation becomes \( x^2 - y^2 = a^2 \).

Step 3: Substitute the given point into the equation

The hyperbola passes through the point \((5,3)\). Substitute \(x = 5\) and \(y = 3\) into the equation \(x^2 - y^2 = a^2\) to find \(a^2\).

Step 4: Calculate \(a^2\) using the point (5, 3)

Substitute into the equation: \(5^2 - 3^2 = a^2\). This simplifies to \(25 - 9 = a^2\), giving \(a^2 = 16\).

Step 5: Write the equation of the hyperbola

Using \(a^2 = 16\), substitute back into the form \(x^2 - y^2 = a^2\) to find the equation: \(x^2 - y^2 = 16\).

Key Concepts

Asymptotes

Asymptotes are crucial lines that help us understand the behavior of a hyperbola as the values of x and y stretch towards infinity. These are straight lines that approach the curve but never actually touch it. In our example, we have asymptotes given by the equations \(y = \pm x\). This means the two lines, \(y = x\) and \(y = -x\), serve as guides for how the hyperbola opens. Asymptotes can reveal important information about the orientation and direction of the hyperbola.

• **Orientation:** If the asymptotes are of the form \(y = \pm \frac{b}{a}x\), the hyperbola is centered at the origin.
• **Direction:** Hyperbolas can open horizontally or vertically, determined by the relationship of x and y in the asymptotes.

Understanding asymptotes allows one to visualize the infinite "wings" of the hyperbola, which stretch towards the asymptotic lines but never intersect them.

Standard Form of Hyperbola

The standard form of a hyperbola is a structured way to express the equation of a hyperbola, giving insight into its geometric properties. For a hyperbola centered at the origin, with asymptotes \(y = \pm \frac{b}{a}x\), the equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This form tells us the lengths and orientation of the axes of the hyperbola.

• **Axes Lengths:** \(a\) and \(b\) determine how far the hyperbola stretches.
• **Equal Axes (\(a = b\)):** When \(a\) equals \(b\), the asymptotes simplify to \(y = \pm x\), as is the case in our exercise.

Converting this equation into a form like \(x^2 - y^2 = a^2\) allows simplified substitution when finding the exact equation of a specific hyperbola.

Substituting Points in Equations

Substituting a point into the equation of a hyperbola is essential to determine specific characteristics, like the size of the hyperbola. By using the point that the hyperbola passes through, such as \((5,3)\) in this exercise, we can calculate \(a^2\), completing the equation.

Here's how it works:

• Substitute \(x = 5\) and \(y = 3\) into \(x^2 - y^2 = a^2\).
• Calculate: \(5^2 - 3^2 = a^2\).
• Solve: \(25 - 9 = a^2\), thus \(a^2 = 16\).

This process locks in the precise shape of the hyperbola and completes its equation. Substitution is a crucial step to achieving an equation that uniquely defines the hyperbola you are dealing with.