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

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

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 \(\frac{x}{y} \propto (x + y)\) 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

8. 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) \(y^2\propto x\)

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

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

11. 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) \(x^2+y^2\)

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

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

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

15. 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.

16. 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.

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