Show that the equation represents a sphere, and find its center and radius. x^{2} + y^{2} + z^{2} + 8x - 6y + 2z + 17 =0

Suman Cole 2020-12-25 Answered
Show that the equation represents a sphere, and find its center and radius.
x2+y2+z2+8x6y+2z+17=0
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Expert Answer

Mayme
Answered 2020-12-26 Author has 103 answers
Consider a sphere with center C(h, k, l) and radius r.
Formula:
Write the expression to find an equation of a sphere with center C (h, k, l) and radius r.
(xh)2+(yk)2+(zl)2=r2(1)
Here,
(h, k, l) is the center of a sphere and
r is the radius of a sphere.
Rearrange the expression x2+y2+z2+8x6y+2z+17=0 as follows.
(x2+8x+4242)+(y26y+3232)+(z2+2z+1212)+17=0
(x2+8x+42)+(y26y+32)+(z2+2z+120+(1691)+17=0
(x+4)2+(y3)2+(z+1)2=2617
(x+4)2+(y3)2+(z+1)2=9
[x(4)]2+(y3)2+[z(1)]2=(3)2(2)
Equation (2) is similar to equation (1).
Therefore, the equation x62+y2+z2+8x6y+2z+17=0 represents a sphere.
Compare equation (2) with equation (1).
h=4
k=3
l=1
r=3
Thus, the center of the spere is (-4, 3, -1) and the radius of the sphere is 3.
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