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 $a$.

You are working as a CAD (Computer-Aided Design) technician for a municipal planning department. You are drafting a new elliptical community park on a coordinate grid. The blueprint specifies that the park's center is shifted away from the origin to the coordinates (h, k), with a horizontal semi-axis of length $a$ and a vertical semi-axis of length $b$. To input the park's boundary correctly into the mapping software, you need to recall how to construct the standard form of the equation: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$.

Arrange the following steps in the correct order to construct this standard form equation starting from the given dimensions.

Custom Elliptical Window Design