You work as a CNC programmer at a custom auto parts manufacturer. You are tasked with machining an elliptical cover plate. The blueprint specifies that the center of the ellipse is shifted to coordinates $(h, k)$ on the machine's reference grid. To verify the software's toolpath, you must recall the standard form of an ellipse equation. Which of the following equations correctly represents this standard form?

A carpenter is using a coordinate system to cut an elliptical opening for a custom kitchen sink. The blueprint specifies that the center of the sink is at $(h, k)$ and the path follows the equation rac{(x-h)^2}{a^2} + rac{(y-k)^2}{b^2} = 1. If the sink is designed to have a vertical major axis, the value of $b$ must be _______ than the value of $a$.

You are working as a layout technician for a custom signage company. You are using a coordinate-based design program to position an elliptical logo on a billboard. The program uses the standard equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. Match each component of the equation or condition with its geometric meaning on the billboard.

Locating the Center of an Elliptical Boundary

A mechanical engineer is designing an elliptical gear defined by the standard equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. True or False: According to the rules for this standard form, if the gear is designed to have a vertical major axis, then the value of $b$ must be greater than the value of $a$.