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

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

7. 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.

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

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

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

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

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

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

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

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

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

17. If \(\left(x^3 - \frac{1}{y^3}\right) \propto \left(x^3 + \frac{1}{y^3}\right)\), then prove that \(xy\) is a constant.

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

19. If \(y\) is the geometric mean between \(x\) and \(z\), then what is the geometric mean between \(x^2 + y^2\) and \(y^2 + z^2\)?

20. 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

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

22. If \(x \propto y\), then – **x is directly proportional to y.** This means: \[ x = ky \quad \text{for some constant } k \] Or, the ratio \(\frac{x}{y}\) remains constant.

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

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

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

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

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

27. If \(x^2 + y^2\) is proportional to \(xy\), prove that \(x+y\) is proportional to \(x-y\).

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

29. If \(x^3 + y^3 \propto x^3 - y^3\), then prove that \(x^2 + xy + y^2 \propto x^2 - xy + y^2\).

30. If \(\cfrac{p}{q} \propto p+q\) and \(\cfrac{q}{p} \propto p-q\), then show that \(p^2 - q^2\) is constant.