(a) Simplify : (frac{x^{2} – 8x + 16}{x^{2} – 7x + 12}).
(b) If (frac{1}{2}, frac{1}{x}, frac{1}{3}) are successive terms of an arithmetic progression (A.P), show that (frac{2 – x}{x – 3} = frac{2}{3}).
Explanation
(a) (frac{x^{2} – 8x + 16}{x^{2} – 7x + 12})
= (frac{(x – 4)(x – 4)}{(x – 4)(x – 3)})
= (frac{(x – 4)}{(x – 3)})
(b) Arithmetic Progression:
(frac{1}{2}, frac{1}{x}, frac{1}{3})
(frac{1}{x} – frac{1}{2} = frac{1}{3} – frac{1}{x})
(frac{2 – x}{2x} = frac{x – 3}{3x})
(frac{2 – x}{x – 3} = frac{2x}{3x})
(implies frac{2 – x}{x – 3} = frac{2}{3})