Question

If `x=cy+bz,y=az+cx,z=bx+ay` then prove that `x^2/(1-a^2)=y^2/(1-b^2)=z^2/(1-c^2)` ?

Answer 1

Please see below.

Explanation:

Let us eliminate `z` from first two equations.

As `x=cy+bz=cy+b(bx+ay)=cy+b^2x+aby`

and multiplying each term by `x`, we get

`x^2=cxy+b^2x^2+abxy` ....................(1)

and similarly `y=az+bx=a(bx+ay)+cx=abx+a^2y+cx`

and multiplying each term by `y`, we get

`y^2=abxy+a^2y^2+cxy` ....................(2)

Subtracting (2) from (1), we have

`x^2-y^2=b^2x^2-a^2y^2`

or `x^2-b^2x^2=y^2-a^2y^2`

or `x^2(1-b^2)=y^2(1-a^2)`

or `x^2/(1-a^2)=y^2/(1-b^2)`

Similarly we can eliminate `x` from equations for `y` and `z` to get

`y^2/(1-b^2)=z^2/(1-c^2)`

and hence `x^2/(1-a^2)=y^2/(1-b^2)=z^2/(1-c^2)`