Three distinct numbers $x, y, z$ form GP in that order and the numbers $7 x + 5 y, 7 y + 5 z, 7 z + 5 x$ form an AP in that order. The common ratio of GP is

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Detailed Solution

Given: Three distinct numbers $x, y, z$ form GP in that order and the numbers $7 x + 5 y, 7 y + 5 z, 7 z + 5 x$ form an AP in that order. The common ratio of GP.

To find: common ratio of G.P.

we have , $x, y, z$ in G.P

let r be the common ratio

Then, $y = x r a n d z = x r^{2}$

Also, $7 x + 5 y, 7 y + 5 z, 7 z + 5 x$ are in A.P.

$\Rightarrow (7 x + 5 z) - (7 x + 5 y) = (7 z + 5 x) - (7 y + 5 x) \Rightarrow 2 y + 5 z - 7 x = 2 z + 5 x - 7 y \Rightarrow 9 y + 3 z - 12 x = 0 \Rightarrow 9 y + 3 z = 12 x \Rightarrow 3 y + z = 4 x \Rightarrow 3 (x r) + {(x r)}^{2} = 4 x (∵ y = x r a n d z = x r^{2}) \Rightarrow 3 r + r^{2} = 4 \Rightarrow r^{2} + 3 r - 4 = 0 \Rightarrow (r + 4) (r - 1) = 0 \Rightarrow r = - 4, 1$