Prove the ratio

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June 30th 2008, 09:48 AM
kalyanram

Prove the ratio

Please prove this...
Given
$x = cy+bz, y = az+cx, z = bx+ay$ ; show that $\frac{x^2}{1-a^2} = \frac{y^2}{1-b^2} = \frac{z^2}{1-c^2}$

~Kalyan
June 30th 2008, 09:50 AM
Moo

Hello,

Quote:

Originally Posted by kalyanram

Please prove this...
Given
$x = cy+bz, y = az+cx, z = bx+ay$ ; show that $\frac{x^2}{1-{a^2}} = \frac{y^2}{1-{b^2}} = \frac{z^2}{1-{c^2}}$

~Kalyan

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

Quote:

Originally Posted by kalyanram

Please prove this...
Given
$x = cy+bz, y = az+cx, z = bx+ay$ ; show that $\frac{x^2}{1-a^2} = \frac{y^2}{1-b^2} = \frac{z^2}{1-c^2}$

~Kalyan

$y = az+cx \Rightarrow y = az+c(cy+bz) = az +cbz + c^2y \Rightarrow y(1 - c^2) = (a+bc)z$ -----------(1)

$z = bx+ay = b(cy+bz) + ay = (bc+a)y+ zb^2 \Rightarrow z(1-b^2) = (a+bc)y$ ----------------------(2)

Divide (1) by (2), to get $\frac{y(1-c^2)}{z(1- b^2)} = \frac{z}{y} \Rightarrow \frac{y^2}{1- b^2} = \frac{z^2}{1 - c^2}$

You can similarly prove the other ratio by eliminating y(or z) in both the equations.

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