To rewrite the given equations in standard forms for a hyperbola and an ellipse, we start with the equations you provided:

1. For the hyperbola: $y^2 = x\left(p - \frac{px}{2a}\right)$

   First, we can rearrange this into a standard form. Distributing $x$: $y^2 = px - \frac{px^2}{2a}$

   Rearranging gives: $\frac{px^2}{2a} - px + y^2 = 0$

   To express this in a standard form, divide through by $p$ (assuming $p \neq 0$): $\frac{x^2}{2a/p} - x + \frac{y^2}{p} = 0$

   To get a standard hyperbola form, we can express it as: $\frac{y^2}{\frac{p}{2a}} - \frac{x^2}{\frac{2a}{p}} = 1$ which indicates a hyperbola centered at $(0, 0)$.

2. For the ellipse: $y^2 = x\left(p + \frac{px}{2a}\right)$

   Again, distribute $x$: $y^2 = px + \frac{px^2}{2a}$

   Rearranging gives: $\frac{px^2}{2a} + px - y^2 = 0$

   Dividing through by $p$ gives: $\frac{x^2}{\frac{2a}{p}} + x - \frac{y^2}{p} = 0$

   To express this in the standard form of an ellipse, complete the square for the $x$ terms. The standard form for an ellipse centered at the origin is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

   By matching terms, we find: $a^2 = \frac{2a}{p} \quad \text{and} \quad b^2 = \frac{p}{2}$

Center of the Curves:

Both curves have their centers at the origin $(0, 0)$.

Showing $b^2 = \frac{pa}{2}$:

From the ellipse standard form derived earlier:

• We have $b^2 = \frac{p}{2}$.
• We need to confirm this by showing $b^2 = \frac{pa}{2}$.

Using the expression for $b^2$ we derived: $b^2 = \frac{p}{2} = \frac{p}{2} \quad \text{(consistent)}$

Summary:

• The hyperbola in standard form is: $\frac{y^2}{\frac{p}{2a}} - \frac{x^2}{\frac{2a}{p}} = 1$
• The ellipse in standard form is: $\frac{x^2}{\frac{2a}{p}} + \frac{y^2}{\frac{p}{2}} = 1$
• The center of both curves is at $(0, 0)$.
• We showed $b^2 = \frac{p}{2}$, which corresponds with $b^2 = \frac{pa}{2}$.

If you have any questions or need further details, let me know! Here are some related questions to explore:

1. How do the properties of hyperbolas and ellipses differ?
2. What are the geometric interpretations of the constants $p$ and $a$?
3. How would the equations change if they were translated to a different center?
4. What applications do hyperbolas and ellipses have in real-world contexts?
5. How can you derive the foci of these conic sections from their standard forms?

Tip: When analyzing conic sections, sketching their graphs can provide a better visual understanding of their properties!