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

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

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

5. If \((u^2 + v^2) \propto (x^2 + y^2)\) and\(uv \propto xy\), then show that\((u + v) \propto (x + y)\) when \(\frac{u}{v} + \frac{v}{u} = \frac{x}{y} + \frac{y}{x}\)."

6. 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.

7. If \(2x^2 + 3y^2 \propto xy\), then show that \(x \propto y\).

8. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that \(x^2 : y^2 = (1 - a^2) : (1 - b^2)\).

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

10. If \(x \propto y\) and \(y \propto z\), then show that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]

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

12. If \(x + y \propto x - y\), then show that \(x^3 + y^3 \propto x^3 - y^3\).

13. If \(x \propto y\) and \(y \propto z\), then show that \(x + y \propto z\).

14. If \(3x - 4y \propto \sqrt{xy}\), then prove that \(x^2 + y^2 \propto xy\).

15. If \(x + y \propto x - y\), then prove that \(x^4 + y^4 \propto x^2 y^2\).

16. If \(x \propto y\), then show that \((x+y) \propto (x-y)\).

17. If \(x \propto y\) and \(y \propto z\), then show that \[ \frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy} \propto \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \]

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

19. If \(x \propto y\) and \(y \propto z\), then prove that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]

20. If \(x \propto y\), \(y \propto z\), and \(z \propto x\), then show that the product of the three non-zero proportionality constants is 1.

21. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that: \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

22. If \(x^2 + y^2 - z^2 \propto x^2 - y^2 + z^2\), then show that \(x^2 \propto (y^2 - z^2)\)

23. If \(\frac{x}{y} \propto (x + y)\) and \(\frac{y}{x} \propto (x - y)\), then find the value of \(x^2 - y^2\).

(a) "Proportional to \(x\)" (b) "Proportional to \(y\)" (c) "Proportional to \(xy\)" (d) constant

24. If \( x \propto yz \) and \( y \propto zx \), then show that \( z \ne 0 \) is a constant.

25. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

26. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} = \frac{z^2}{1 - c^2} \]

27. If \(x + y \propto x - y\), then prove that \(x^3 + y^3 \propto x^3 - y^3\).

28. If \(\frac{1}{x^3} + \frac{1}{y^3} \propto \frac{1}{x^3 + y^3}\), then show that \(x \propto y\).

29. If \(x \propto z\) and \(y \propto z\), then \(x^2 + y^2 \propto z^2\).

30. If \(x+y \propto x-y\), then prove that \(x^3+y^3\propto xy(x+y)\).