Question

Find the relationship between a, b and c if 5^a = 3^b = 225^c. Where a is not = b, b is not = c and c is not = a.
(A) 1/c = 1/a + 1/b
(B) 1/2c = ab/a+b
(C) c = ab/2(a + b)
(D) c/2 = 1/a + 1/b

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\displaystyle\;2^a=3^b=225^c}$

$\underline{\textbf{To find:}}$

$\textsf{The relation between a,b and c}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\displaystyle\;5^a=3^b=225^c=k(say)}$

$\mathsf{\displaystyle\;5^a=k\;\implies\;5=k^\frac{1}{a}}$

$\mathsf{\displaystyle\;3^b=k\;\implies\;3=k^\frac{1}{b}}$

$\mathsf{\displaystyle\;225^c=k\;\implies\;225=k^\frac{1}{c}}$

$\mathsf{Now,}$

$\mathsf{\displaystyle\;225=k^\frac{1}{c}}$

$\mathsf{\displaystyle\;15^2=k^\frac{1}{c}}$

$\mathsf{\displaystyle\;(5{\times}3)^2=k^\frac{1}{c}}$

$\mathsf{\displaystyle\;(k^\frac{1}{a}{\times}k^\frac{1}{b})^2=k^\frac{1}{c}}$

$\mathsf{\displaystyle\;\left(k^{\frac{1}{a}+\frac{1}{b}}\right)^2=k^\frac{1}{c}}$

$\mathsf{\displaystyle\;\left(k^{\frac{b+a}{ab}}\right)^2=k^\frac{1}{c}}$

$\mathsf{\displaystyle\;k^{\frac{2(a+b)}{ab}}=k^\frac{1}{c}}$

$\textsf{Equating powers on bothsides, we get}$

$\mathsf{\dfrac{2(a+b)}{ab}=\dfrac{1}{c}}$

$\textsf{Taking reciprocals, we get}$

$\mathsf{\dfrac{ab}{2(a+b)}=c}$

$\boxed{\mathsf{c=\dfrac{ab}{2(a+b)}}}$

$\underline{\textbf{Answer:}}$

$\textsf{Option (c) is correct }$