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

Question

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

✓

Answer

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

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