Find the :
(i) fourth proportional to `2xy`, `x^(2)` and `y^(2)`.
(ii) third proportional to `a^(2) - b^(2)` and `a + b`.
(iii) mean proportion to `(x - y)` and `(x^(3) - x^(2)y)`

Answer

Let's solve the given problems step by step.

### (i) Fourth Proportional to `2xy`, `x^2`, and `y^2`

To find the fourth proportional, we use the relationship:

\[ \frac{a}{b} = \frac{c}{d} \]

Where \( a = 2xy \), \( b = x^2 \), and \( c = y^2 \). We need to find \( d \).

1. Set up the equation:
\[ \frac{2xy}{x^2} = \frac{y^2}{d} \]

2. Cross-multiply:
\[ 2xy \cdot d = y^2 \cdot x^2 \]

3. Solve for \( d \):
\[ d = \frac{y^2 \cdot x^2}{2xy} \]

4. Simplify:
\[ d = \frac{xy}{2} \]

Thus, the fourth proportional is \( \frac{xy}{2} \).

### (ii) Third Proportional to `a^2 - b^2` and `a + b`

To find the third proportional, we use the relationship:

\[ \frac{a}{b} = \frac{b}{c} \]

Where \( a = a^2 - b^2 \), \( b = a + b \), and we need to find \( c \).

1. Set up the equation:
\[ \frac{a^2 - b^2}{a + b} = \frac{a + b}{c} \]

2. Cross-multiply:
\[ (a^2 - b^2) \cdot c = (a + b) \cdot (a + b) \]

3. Recognize that \( a^2 - b^2 = (a + b)(a - b) \):
\[ (a + b)(a - b) \cdot c = (a + b)^2 \]

4. Cancel \( a + b \) (assuming \( a + b \neq 0 \)):
\[ (a - b) \cdot c = a + b \]

5. Solve for \( c \):
\[ c = \frac{a + b}{a - b} \]

Thus, the third proportional is \( \frac{a + b}{a - b} \).

### (iii) Mean Proportion to `x - y` and `x^3 - x^2y`

To find the mean proportion, we use the relationship:

\[ \frac{a}{c} = \frac{c}{b} \]

Where \( a = x - y \), \( b = x^3 - x^2y \), and we need to find \( c \).

1. Set up the equation:
\[ \frac{x - y}{c} = \frac{c}{x^3 - x^2y} \]

2. Cross-multiply:
\[ (x - y) \cdot (x^3 - x^2y) = c^2 \]

3. Factor out \( x^2 \) from \( x^3 - x^2y \):
\[ x^3 - x^2y = x^2(x - y) \]

4. Substitute back into the equation:
\[ (x - y) \cdot x^2(x - y) = c^2 \]

5. Simplify:
\[ c^2 = x^2(x - y)^2 \]

6. Take the square root:
\[ c = x(x - y) \]

Thus, the mean proportion is \( x(x - y) \).

### Summary of Answers:
1. Fourth Proportional: \( \frac{xy}{2} \)
2. Third Proportional: \( \frac{a + b}{a - b} \)
3. Mean Proportion: \( x(x - y) \)