Question
If (pa + qb) : (pc + qd) :: (pa – qb) : (pc – qd) prove that a : b : : c : d
✓
Answer
$
\begin{aligned}
& (p a+q b):(p c+q d)::(p a-q b):(p c-q d) \\
& \Rightarrow \frac{p a+q b}{p c+q d}=\frac{p q-q b}{p c-q d} \\
& \Rightarrow \frac{p a+q b}{p c-q d}=\frac{p q+q b}{p c-q d}
\end{aligned}$
Applying componendo and dividendo
$
\begin{aligned}
& \Rightarrow \frac{ pa + qb + pa - qb }{ pa + qb - pa + qb }=\frac{ pc + qs + pc - qd }{ pc - qd - pc + qd } \\
& \Rightarrow \frac{2 p a}{2 q b}=\frac{2 p c}{2 q d} \\
& \Rightarrow \frac{a}{b}=\frac{c}{d} \quad \ldots\left(\text { Dividing by } \frac{2 p}{2 q}\right)
\end{aligned}
$
Hence $a : b :: c = d$.
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