If x/a=y/b=z/c .Show that (x2+y2+z2 )/(a2+b2+c2 ) =[(px+qy+rz)/(pa+qb+rc)]2
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Answer:
(a
2
+b
2
+c
2
)(x
2
+y
2
+z
2
)=(ax+by+cz)
2
a
2
x
2
+a
2
y
2
+a
2
z
2
+b
2
x
2
+b
2
y
2
+b
2
z
2
+c
2
x
2
+c
2
y
2
+c
2
z
2
=a
2
x
2
+b
2
y
2
+c
2
z
2
+2abxy+2bcyz+2aczx
Rearranging the terms
b
2
x
2
+a
2
y
2
−2abxy+c
2
y
2
+b
2
z
2
−2bcyz+c
2
x
2
+a
2
z
2
−2aczx=0
(bx−ay)
2
+(cy−bz)
2
+(cx−az)
2
=0
⇒bx−ay=0,cy−bz=0,cx−az=0
⇒
a
x
=
b
y
,
b
y
=
c
z
,
c
z
=
a
x
⇒
a
x
=
b
y
=
c
z
hence proved.
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