Question

if x=cy+bz, y=az+cx, z=bx+ay, then prove that x^2:y^2=1-a^2:1-b^2.​

Answer 1

Let us eliminate
z
from first two equations.
As
x
=
c
y
+
b
z
=
c
y
+
b
(
b
x
+
a
y
)
=
c
y
+
b
2
x
+
a
b
y
and multiplying each term by
x
, we get
x
2
=
c
x
y
+
b
2
x
2
+
a
b
x
y
....................(1)
and similarly
y
=
a
z
+
b
x
=
a
(
b
x
+
a
y
)
+
c
x
=
a
b
x
+
a
2
y
+
c
x
and multiplying each term by
y
, we get
y
2
=
a
b
x
y
+
a
2
y
2
+
c
x
y
....................(2)
Subtracting (2) from (1), we have
x
2
−
y
2
=
b
2
x
2
−
a
2
y
2
or
x
2
−
b
2
x
2
=
y
2
−
a
2
y
2
or
x
2
(
1
−
b
2
)
=
y
2
(
1
−
a
2
)
or
x
2
1
−
a
2
=
y
2
1
−
b
2