find the equation of the sphere through the four points and determine its radius (0,0,0) (a,0,0) (0,b,0) (0,0,c)
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Explanation:
Since the sphere passes through origin, its equation is of the form
x
2
+y
2
+z
2
+2ux+2vy+2wz=0. ...(1)
Substituting the co - ordinates of the other given points, we get
a
2
+b
2
+c
2
−2ua+2vb+2wc=0. ...(2)
a
2
+b
2
+c
2
−2ua−2vb+2wc=0. ...(3)
a
2
+b
2
+c
2
+2ua+2vb−2wc=0. ...(4)
Adding (2) and (3), 2(a
2
+b
2
+c
2
)=−4wc
2w=
−b
a
2
+b
2
+c
.
Similarly, 2v=
−c
a
2
+b
2
+c
and 2v=
−a
a
2
+b
2
+c
Putting the values of u, v, w in (1), we get
x
2
+y
2
+z
2
−(a
2
+b
2
+c
2
)(
a
x
+
b
y
+
c
z
)=0.
The radius of the sphere is
(u
2
+v
2
+w
2
)
,d being zero
R=
{
4
(a
2
+b
2
+c
2
)
2
(
a
2
1
+
b
2
1
+
c
2
1
)}
=
2
a
2
+b
2
+c
2
(a
−2
+b
−2
+c
−2
)
.
Here a=1,b=1,c=1. Thus radius of the sphere is R=
2
3
3
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