Let's tackle the equations one at a time. I’ll first reduce them to their standard forms, then identify the hyperbola's center, vertices, foci, latera recta, and directrix. For clarity, let’s proceed with part (a) first:

Part (a): $4x^2 - 9y^2 + 24x + 108y - 324 = 0$

1. Group terms for $x$ and $y$:
   $4x^2 + 24x - 9y^2 + 108y = 324$

2. Factor coefficients:
   $4(x^2 + 6x) - 9(y^2 - 12y) = 324$

3. Complete the square for $x$ and $y$:

   • For $x^2 + 6x$, add and subtract $(6/2)^2 = 9$.
   • For $y^2 - 12y$, add and subtract $(-12/2)^2 = 36$.

   Rewrite: [ 4[(x + 3)^2 - 9] - 9[(y - 6)^2 - 36] = 324 ]

4. Simplify:
   $4(x + 3)^2 - 36 - 9(y - 6)^2 + 324 = 324$
   $4(x + 3)^2 - 9(y - 6)^2 = 36$

5. Divide by 36 to get the standard form: $\frac{(x + 3)^2}{9} - \frac{(y - 6)^2}{4} = 1$

This is a hyperbola with:

• Center: $(-3, 6)$
• Vertices: $(-3 \pm 3, 6)$ → $(-6, 6)$ and $(0, 6)$
• Foci: Found using $c = \sqrt{a^2 + b^2} = \sqrt{9 + 4} = \sqrt{13}$, so foci are $(-3 \pm \sqrt{13}, 6)$.
• Transverse axis: Horizontal
• Latus rectum: Length $2b^2/a = 8/3$.

Would you like to delve into parts (b) and (c) now? Let me know!

Related Questions:

1. How do you identify a hyperbola's transverse and conjugate axes?
2. Why is completing the square important for conic sections?
3. Can you derive the equations for the asymptotes of a hyperbola?
4. What are the differences between ellipse and hyperbola properties?
5. How does the orientation of a hyperbola change with the terms $x^2$ and $y^2$?

Tip:

Always verify whether the hyperbola is vertical or horizontal by observing the terms' coefficients and signs in the standard form!