The exercise provided contains problems on hyperbolas. I'll break down the tasks step by step.

Problem 1

Reduce the given equations to standard form and draw the hyperbolas:

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

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

(b) $16x^2 - 9y^2 - 96x - 144y - 288 = 0$

1. Follow a similar process to part (a), grouping and completing squares.

(c) $25x^2 - 64y^2 + 1600 = 0$

1. Rearrange to isolate terms and bring to standard form.

Problem 2

Equation of a hyperbola centered at $(-4, 2)$:

• Transverse axis: parallel to $y$-axis.
• Eccentricity: $\frac{\sqrt{61}}{5}$.
• Distance between directrices: $\frac{50\sqrt{61}}{61}$.

Start with the standard form of a hyperbola with vertical transverse axis: $\frac{(x+4)^2}{b^2} - \frac{(y-2)^2}{a^2} = 1$ Use the given eccentricity and directrices formula to find $a^2$ and $b^2$.

Problem 3

Equation of a hyperbola centered at the origin with:

• Transverse axis along $x$-axis.
• Length of the latus rectum $= 1$.
• Asymptotes $y = \pm \frac{1}{2}x$.

Use:

1. Equation of hyperbola with horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
2. Latus rectum formula: $\frac{2b^2}{a}$.
3. Slope of asymptotes: $y = \pm \frac{b}{a}x$.

Would you like me to work through one or all of these problems with detailed steps?

5 Questions to Explore:

1. How do you determine the standard form of a hyperbola?
2. What is the role of eccentricity in identifying the shape of a hyperbola?
3. How does completing the square help in transforming equations?
4. What is the relationship between asymptotes and the hyperbola's axes?
5. How are latus rectum and directrix connected to hyperbola parameters?

Tip:

Always check if the hyperbola opens horizontally or vertically based on its general equation, as it determines the placement of $a^2$ and $b^2$.