Question

If 3^a=5^b=15^c(a,b,c is not =0)then 1/c^2-(1/a^2+1/b^2)=

Answer 1

Let 3^a = 5^b = 15^c = k

$3^a = k then 3 = k^{ \frac{1}{a} }$
$5^b = k then 5 = k^{ \frac{1}{b} }$
$15^c = k then 15 = k^{ \frac{1}{c} }$

$15 = k^{ \frac{1}{c} }$
$5*3 = k^{ \frac{1}{c} }$
$k^{ \frac{1}{b} * k^{ \frac{1}{a} = k^{ \frac{1}{c} }$
$k^{ \frac{1}{b} + { \frac{1}{a} = k^{ \frac{1}{c} }$

${ \frac{1}{b}+{ \frac{1}{a} = { \frac{1}{c} }$

Squaring on both sides

$({ \frac{1}{b}+{ \frac{1}{a})^{2} = ({ \frac{1}{c} })^2$

$\frac{1}{a^2} + \frac{1}{b^2} + 2. \frac{1}{a}. \frac{1}{a} = \frac{1}{c^2}$

$\frac{2}{ab} = \frac{1}{c^2} - ( \frac{1}{a^2} + \frac{1}{b^2} )$