# Prove the ratio

• Jun 30th 2008, 09:48 AM
kalyanram
Prove the ratio
Given
$\displaystyle x = cy+bz, y = az+cx, z = bx+ay$ ; show that $\displaystyle \frac{x^2}{1-a^2} = \frac{y^2}{1-b^2} = \frac{z^2}{1-c^2}$

~Kalyan
• Jun 30th 2008, 09:50 AM
Moo
Hello,

Quote:

Originally Posted by kalyanram
Given
$\displaystyle x = cy+bz, y = az+cx, z = bx+ay$ ; show that $\displaystyle \frac{x^2}{1-{a^2}} = \frac{y^2}{1-{b^2}} = \frac{z^2}{1-{c^2}}$

~Kalyan

The syntax is \frac{...}{...} :)
• Jun 30th 2008, 10:01 AM
Isomorphism
Quote:

Originally Posted by kalyanram
$\displaystyle x = cy+bz, y = az+cx, z = bx+ay$ ; show that $\displaystyle \frac{x^2}{1-a^2} = \frac{y^2}{1-b^2} = \frac{z^2}{1-c^2}$
$\displaystyle y = az+cx \Rightarrow y = az+c(cy+bz) = az +cbz + c^2y \Rightarrow y(1 - c^2) = (a+bc)z$-----------(1)
$\displaystyle z = bx+ay = b(cy+bz) + ay = (bc+a)y+ zb^2 \Rightarrow z(1-b^2) = (a+bc)y$----------------------(2)
Divide (1) by (2), to get $\displaystyle \frac{y(1-c^2)}{z(1- b^2)} = \frac{z}{y} \Rightarrow \frac{y^2}{1- b^2} = \frac{z^2}{1 - c^2}$