In your role as a CAD drafter for an engineering firm, you are modeling a hyperbolic cooling tower. The blueprints require a hyperbola that opens left and right, with its center shifted from the origin to a specific coordinate (h, k) on your site grid. To correctly program this structural shape into the design software, which standard form equation must you use?

As a technician for a city planning department, you are using hyperbolic curves to design the seating layout of a new urban park's amphitheater. To ensure the design software correctly interprets your site map, match the following mathematical components of the hyperbola's standard form equation with their geometric descriptions on the layout.

Locating the Center of a Hyperbolic Boundary

A site supervisor is reviewing blueprints for a hyperbolic cooling tower. The cross-section is modeled by the equation $\frac{(y - 10)^2}{64} - \frac{(x - 15)^2}{36} = 1$. True or False: According to the standard form of the equation, the center of this hyperbolic shape is located at (10, 15).

A city planner is using a hyperbolic curve to design the layout of a new urban plaza. The curve is centered at the point (12, 18) on the city's master grid. To correctly model this shape using the standard form equation $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$, the planner must identify the correct value for $k$. Based on the given center, the numerical value that should be substituted for $k$ is ____.

You are a landscape design assistant preparing a blueprint for a decorative courtyard with a pathway shaped like a horizontal hyperbola. The pathway is modeled by the standard equation $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$. To help the construction team stake out the exact layout, you need to first draw the central guide rectangle on your grid map.

Arrange the steps in the correct order to construct this central guide rectangle on your coordinate grid, starting from the given equation.

Onboarding Reference Guide: Standard Form of a Hyperbola with Center (h, k)