What is the standard form equation for the asymptotes of a horizontal hyperbola?

What is the standard form equation for the asymptotes of a horizontal hyperbola?

• A. (x - h)²/a² - (y - k)²/b² = 0
• B. (y - k)²/a² - (x - h)²/b² = 1
• C. (x - h)²/a² - (y - k)²/b² = 1
• D. (x + h)²/b² - (y + k)²/a² = 1

Answer: (C)

The standard form equation for a hyperbola involves terms that demonstrate how the shape is oriented in the coordinate system. In the case of a horizontal hyperbola, its standard form is expressed as \((x - h)²/a² - (y - k)²/b² = 1\). Here, \((h, k)\) represents the center of the hyperbola, while \(a\) is linked to the distance from the center to the vertices, and \(b\) relates to the distance from the center to the asymptotes. The difference in signs indicates that hyperbolas consist of two branches that open outward. As for the asymptotes, these lines can be derived from this equation, resulting in two linear equations: \(y = k \pm \frac{b}{a}(x - h)\). Upon examining this context, the choice that accurately depicts the standard form of a horizontal hyperbola is indeed presented in the correct answer, where the negative sign in the second term ensures the proper orientation and nature of the hyperbola.

The standard form equation for a hyperbola involves terms that demonstrate how the shape is oriented in the coordinate system. In the case of a horizontal hyperbola, its standard form is expressed as ((x - h)²/a² - (y - k)²/b² = 1).

Here, ((h, k)) represents the center of the hyperbola, while (a) is linked to the distance from the center to the vertices, and (b) relates to the distance from the center to the asymptotes. The difference in signs indicates that hyperbolas consist of two branches that open outward.

As for the asymptotes, these lines can be derived from this equation, resulting in two linear equations: (y = k \pm \frac{b}{a}(x - h)). Upon examining this context, the choice that accurately depicts the standard form of a horizontal hyperbola is indeed presented in the correct answer, where the negative sign in the second term ensures the proper orientation and nature of the hyperbola.