(x/a) + (y/b) + (z/c)=√2
(a/x) + (b/y) + (c/z)=0
Find (x²/a²) + (y²/b²) + (z²/c²)
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Explanation:
Suppose you have three points in
R
3
p
1
=
(
a
,
b
,
c
)
p
2
=
(
x
,
y
,
z
)
p
0
=
(
0
,
0
,
0
)
The respective lengths of sides
p
1
−
p
0
and
p
2
−
p
0
are
∥
p
1
−
p
0
∥
=
√
a
2
+
b
2
+
c
2
∥
p
2
−
p
0
∥
=
√
x
2
+
y
2
+
z
2
and their scalar product
(
p
1
−
p
0
)
.
(
p
2
−
p
0
)
=
a
x
+
b
y
+
c
z
but
(
p
1
−
p
0
)
.
(
p
2
−
p
0
)
=
∥
p
1
−
p
0
∥
∥
p
2
−
p
0
∥
cos
(
ˆ
p
1
p
0
p
2
)
so
cos
(
ˆ
p
1
p
0
p
2
)
=
a
x
+
b
y
+
c
z
√
a
2
+
b
2
+
c
2
√
x
2
+
y
2
+
z
2
=
30
5
×
6
=
1
Then the sides
p
1
−
p
0
and
p
2
−
p
0
are aligned so
(
p
1
−
p
0
)
=
λ
(
p
2
−
p
0
)
and of course
λ
=
5
6
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