1. If \(x : a = y : b = z : c\), then prove that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)

2. If \(x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1\), then show that \(\cfrac{a}{a + x} + \cfrac{b}{b + y} + \cfrac{c}{c + z} = 1\).

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

4. If \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] then show that \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

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

6. If \(x : a = y : b = z : c\), then show that \(\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}\).

7. Given: \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] Show that: \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

8. If \(x : a = y : b = z : c\), then prove that \(\cfrac{x^3}{a^2} + \cfrac{y^3}{b^2} + \cfrac{z^3}{c^2} = \cfrac{(x + y + z)^3}{(a + b + c)^2}\)

9. If \(x : a = y : b = z : c\), then show that \(\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\).

10. If \(x : a = y : b = z : c\), then show that \[ \frac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \frac{xyz}{abc} \]

11. If \(x : a = y : b = z : c\), then prove that \(\cfrac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \cfrac{xyz}{abc}\)

12. If \(x : a = y : b = z : c\), then prove that \[\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\]

13. If \((a^2 + b^2)(x^2 + y^2) = (ax + by)^2\), then what are the values of \(x\) and \(y\)?

(a) \(b:a\) (b) \(b^2:a^2\) (c) \(a^2:b^2\) (d) \(a:b\)

14. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

15. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

16. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1:r\), then show that \((r+1)^2ac = b^2r\).

17. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

18. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

19. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

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

21. If the roots of the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), are in the ratio 1:2, then show that \(2b^2 = 9ac\).

22. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be:

(a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13

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

24. Given: \(\frac{x + y}{x - y} = \frac{a}{b}\) Prove that: \(\frac{y^2 + xy}{x^2 - xy} = \frac{a^2 - ab}{b^2 + ab}\)

25. If \(a : b = b : c\), then show that \[abc (a + b + c)^3 = (ab + bc + ca)^3\]

26. If \(a : b = c : d\), then prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

27. If \(a : b = b : c\), then prove that \((a + b + c)(a - b + c) = a^2 + b^2 + c^2\).

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

29. If \(a : b = b : c\), then prove that \[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

30. If \(a:b = b:c\), show that \((a+b+c)(a-b+c) = a^2 + b^2 + c^2\).