Consider the following for the next items that follow: Let \(\rm \frac{(x-a)(x-b)}{(x-m a)(x-m b)}=\frac{(x+a)(x+b)}{(x+m a)(x+m b)}\); m, a, b > 0.
What is x equal to ?
Concept:
Componendo Dividendo Rule:
If (a + b ) : (a – b) = (c + d) : (c – d), than
a : b = c : d
Calculation:
According to the question,
\(\rm \frac{(x-a)(x-b)}{(x-m a)(x-m b)}=\frac{(x+a)(x+b)}{(x+m a)(x+m b)}\)
⇒ \((x - a) (x - b) \over (x + a) (x + b) \) = \((x - ma) (x - mb) \over (x + ma) (x + mb)\)
⇒ \(x^2 - (a + b)x + ab \over x^2 + (a + b)x + ab\) = \(x^2 - (ma + mb)x + m^2ab \over x^2 + (ma + mb)x + m^2ab \)
By componendo dividendo method,
⇒ \(x^2 + ab \over (a + b)x \) = \(x^2 + m^2 ab \over (ma + mb)x \)
⇒ \(x^2 + ab \over x^2 + m^2 ab\) = \((a + b) \over (a + b) m\)
⇒ \(x^2 + ab \over x^2 + m^2 ab\) = \(1 \over m\)
⇒ mx2 + mab = x2 + m2ab
⇒ mx2 - x2 = m2ab - mab
⇒ x2 (m - 1) = mab (m - 1)
⇒ x2 = mab
⇒ x = \(\pm \sqrt{\mathrm{mab}}\)
∴ The value of x is equal to \(\pm \sqrt{\mathrm{mab}}\).
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