Let's analyze and solve each part of the question.

Part (a): $9y^2 = 144 + 16x^2$

Step 1: Rearrange the equation

We rewrite the given equation in standard form for a conic section: $9y^2 - 16x^2 = 144$

Step 2: Divide by 144

To simplify, divide the entire equation by 144: $\frac{9y^2}{144} - \frac{16x^2}{144} = \frac{144}{144}$

Simplify each term: $\frac{y^2}{16} - \frac{x^2}{9} = 1$

Step 3: Identify the conic section

This equation is in the standard form of a hyperbola: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ where $a^2 = 16$ and $b^2 = 9$.

Answer for (a):

• Equation: $\frac{y^2}{16} - \frac{x^2}{9} = 1$
• Type: Hyperbola

Part (b): $5y^2 = 25 - 5x^2$

Step 1: Rearrange the equation

Rewrite the given equation: $5y^2 + 5x^2 = 25$

Step 2: Divide by 25

Divide the entire equation by 25 to normalize it: $\frac{5y^2}{25} + \frac{5x^2}{25} = \frac{25}{25}$

Simplify each term: $\frac{y^2}{5} + \frac{x^2}{5} = 1$

Step 3: Identify the conic section

This equation is in the standard form of a circle: $\frac{x^2}{r^2} + \frac{y^2}{r^2} = 1$ where $r^2 = 5$.

Answer for (b):

• Equation: $\frac{y^2}{5} + \frac{x^2}{5} = 1$
• Type: Circle

Do you need further details or examples? Here are five follow-up questions to consider:

1. How do you distinguish between a hyperbola and an ellipse in standard form?
2. What would happen if the coefficients of $x^2$ and $y^2$ were the same in part (b)?
3. How do we determine the center of the conic sections from the equations?
4. How would the equations change if the hyperbola in part (a) were rotated?
5. What is the geometric significance of $a$ and $b$ in the hyperbola equation?

Tip: Always simplify equations by dividing by the constant term to express them in standard form.