A machinist is using a CNC router to carve a hyperbolic design defined by the equation ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$. After following the procedure to rewrite this in standard form, which coordinates should be used for the center (h, k) of the hyperbola?

A structural engineer is analyzing a hyperbolic support beam defined by the equation ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$. To determine the beam's geometric properties, the engineer must convert this general equation into its standard form. Arrange the following steps in the correct order to complete the conversion.

An architectural designer is analyzing a hyperbolic arch defined by the equation ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$. When rewritten in standard form, the equation is $\frac{(x - 3)^2}{9} - \frac{(y + 2)^2}{4} = 1$. To prepare for a structural report, the designer needs to identify the geometric properties of this hyperbola. Match each geometric property to its correct value or description.

A technician is mapping a hyperbolic path defined by the general equation ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$. After converting the equation to its standard form, $\frac{(x - 3)^2}{9} - \frac{(y + 2)^2}{4} = 1$, the technician identifies the center of the hyperbola at (3, -2). Since the $x^2$ term is positive in the standard form, the technician concludes that the hyperbola opens horizontally and that both vertices will also have a $y$-coordinate of -2. Is this statement true or false?

A structural drafter is converting the general equation of a hyperbolic arch, ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$, into standard form to determine its physical dimensions. During the grouping step, the drafter factors the terms to yield ${}4(x^2 - 6x) - 9(y^2 + 4y) = 36$. To complete the square for the $y$-terms inside the parentheses, the drafter must add the number ____.

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