If (3a + 8b) : (3c + 8d) = (3a − 8b) : (3c − 8d), then show that a, b, c, d are in proportion.

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Solution

$Given : \frac{3 a + 8 b}{3 c + 8 d} = \frac{3 a - 8 b}{3 c - 8 d} T h u s, w e h a v e : (3 a + 8 b) (3 c - 8 d) = (3 a - 8 b) (3 c + 8 d) \Rightarrow 9 a c - 24 a d + 24 b c - 64 b d = 9 a c + 24 a d - 24 b c - 64 b d \Rightarrow - 24 a d - 24 a d = - 24 b c - 24 b c \Rightarrow - 48 a d = - 48 b c \Rightarrow a d = b c \Rightarrow \frac{a}{b} = \frac{c}{d}$

Hence, a, b, c and d are in proportion.

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