Question

If $\frac{a}{(a+b)(a+c)}=\frac{1}{a+b+c+d}$ then show that a, b, c, d are in proportion.

Answer 1

$$\begin{gathered} \frac{a}{\left(a+b\right)\left(a+c\right)}=\frac{1}{a+b+c+d} \\ \mathrm{On}\mathrm{cross}-\mathrm{multiplying},\mathrm{we}\mathrm{get}: \\ a\left(a+b+c+d\right)=\left(a+b\right)\left(a+c\right) \\ \Rightarrow a^2+ab+ac+ad=a^2+ac+ba+bc \\ \Rightarrow a^2+ab+ac+ad-a^2-ac-ba-bc=0 \\ \Rightarrow ad-bc=0 \\ \Rightarrow ad=bc \\ \Rightarrow \frac{a}{b}=\frac{c}{d} \\ \mathrm{Hence},a,b,c\mathrm{and}d\mathrm{are}\mathrm{in}\mathrm{proportion}. \end{gathered}$$