If a, b, c and d are in proportion, prove that: a^2+a b c^2+c d =… - Vidyadip

Question

If $a, b, c$ and $d$ are in proportion, prove that:
$
\frac{a^2+a b}{c^2+c d}=\frac{b^2-2 a b}{d^2-2 c d}
$

✓

Answer

$\begin{aligned} & \because a , b , c , d \text { are in proportion } \\ & \frac{a}{b}=\frac{c}{d}= k \text { (say) } \\ & a = bk , c = dk . \\ & \text { L.H.S. }=\frac{a^2+a b}{c^2+c d} \\ & =\frac{k^2 b^2+b k . b}{k^2 d^2+d k \cdot d} \\ & =\frac{k b^2(k+1)}{d^2 k(k+1)} \\ & =\frac{b^2}{d^2} \\ & \text { R.H.S. }=\frac{b^2-2 a b}{d^2-2 c d} \\ & =\frac{b^2-2 \cdot b k b}{d^2-2 d k d} \\ & =\frac{b^2(1-2 k)}{d^2(1-2 k)} \\ & =\frac{b^2}{d^2} \\ & \therefore \text { L.H.S. }=\text { R.H.S. }\end{aligned}$

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