Question

Given four quantities a, b, c and d are in proportion. Show that: (a – c)b² : (b – d)cd = (a² – b² – ab) : (c² – d² – cd)

Answer 1

Let ${\displaystyle \frac{a}{b}=\frac{c}{d}=k}$

${\displaystyle \Rightarrow }$ a = bk and c = dk

L.H.S = ${\displaystyle \frac{(a-c)b^2}{(b-d)cd}}$

= ${\displaystyle \frac{(bk-dk)b^2}{(b-d)d^{2}k}}$

= ${\displaystyle \frac{b^2}{d^2}}$

R.H.S = ${\displaystyle \frac{a^2-b^2-ab}{c^2-d^2-cd}}$

= ${\displaystyle \frac{b^2{k}^2-b^2-bkb}{d^2{k}^2-d^2- dkd}}$

= ${\displaystyle \frac{b^2(k^2-1-k)}{d^2(k^2-1-k)}}$

= ${\displaystyle \frac{b^2}{d^2}}$

${\displaystyle \Rightarrow }$ L.H.S = R.H.S

Hence proved.