1. If \(\cfrac{x}{y} \propto (x + y)\) and \(\cfrac{y}{x} \propto (x - y)\), then show that \((x^2 - y^2)\) is constant.

2. If \(x ∝ y\) and \(y ∝ z\), then prove that \((x^2 + y^2 + z^2) ∝ (xy + yz + zx)\).

3. If \(\left(\frac{1}{x} - \frac{1}{y}\right) \propto \frac{1}{x - y}\), then show that \((x^2 + y^2) \propto xy\).

4. What is the fourth proportional of \((x + y)\), \((x^2 - y^2)\), and \((x^2 + xy + y^2)\)?

(a) \(x^3-y^3\) (b) \(x^3+y^3\) (c) \(x^2-y^2\) (d) \(x^2+y^2\)

5. If \((x^2 + y^2) \propto (x^2 - y^2)\), then which of the following is correct?

(a) \(x\propto y\) (b) \(xy=\) constant (c) \(x^2\propto y\) (d) \(y^2\propto x\)

6. Find the fourth proportional of \((x^2 - y^2), (x^2y - xy^2), (x + y)\).

7. If \(\frac{x}{y} \propto x + y\) and \(\frac{y}{x} \propto x - y\), then show that \((x^2 - y^2)\) is a constant quantity.

8. If \(x + y ∝ x - y\), then show that \(x^2 + y^2 ∝ xy\).

9. If \(x : y = (a + 2) : (a - 2)\), then show that \((x^2 + y^2) : (x^2 - y^2) = 4a : (a^2 + 4)\).