If a, b, c, d are in G.P., prove that: (b+c)(b+d)=(c+a)(c+d) - Vidyadip

Question

If a, b, c, d are in G.P., prove that: $(\text{b}+\text{c})(\text{b}+\text{d})=(\text{c}+\text{a})(\text{c}+\text{d})$

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Answer

a, b and c are in G.P.$\therefore\text{b}^2=\text{ac }\cdots(1)$
$\text{L.H.S}=({\text{b}+\text{a})(\text{b}+\text{d})}$ $=\text{b}^2+\text{bd}+\text{bc}+\text{cd}$ $=\text{ac}+\text{c}^2+\text{ad}+\text{cd}$ $[\text{Using (1)}]$ $=\text{c}(\text{a}+\text{c})+\text{d}(\text{a}+\text{c})$ $=(\text{c}+\text{a})(\text{c}+\text{d})$ $=\text{R.H.S}$$\therefore\text{R.H.S}=\text{L.H.S}$

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