Suman Cole

2020-12-25

Show that the equation represents a sphere, and find its center and radius.

${x}^{2}+{y}^{2}+{z}^{2}+8x-6y+2z+17=0$

Mayme

Skilled2020-12-26Added 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.

$(x-h{)}^{2}+(y-k{)}^{2}+(z-l{)}^{2}={r}^{2}(1)$

Here,

(h, k, l) is the center of a sphere and

r is the radius of a sphere.

Rearrange the expression${x}^{2}+{y}^{2}+{z}^{2}+8x-6y+2z+17=0$ as follows.

$({x}^{2}+8x+{4}^{2}-{4}^{2})+({y}^{2}-6y+{3}^{2}-{3}^{2})+({z}^{2}+2z+{1}^{2}-{1}^{2})+17=0$

$({x}^{2}+8x+{4}^{2})+({y}^{2}-6y+{3}^{2})+({z}^{2}+2z+{1}^{20}+(-16-9-1)+17=0$

$(x+4{)}^{2}+(y-3{)}^{2}+(z+1{)}^{2}=26-17$

$(x+4{)}^{2}+(y-3{)}^{2}+(z+1{)}^{2}=9$

$[x-(-4){]}^{2}+(y-3{)}^{2}+[z-(-1){]}^{2}=(3{)}^{2}(2)$

Equation (2) is similar to equation (1).

Therefore, the equation$x62+{y}^{2}+{z}^{2}+8x-6y+2z+17=0$ 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.

Formula:

Write the expression to find an equation of a sphere with center C (h, k, l) and radius r.

Here,

(h, k, l) is the center of a sphere and

r is the radius of a sphere.

Rearrange the expression

Equation (2) is similar to equation (1).

Therefore, the equation

Compare equation (2) with equation (1).

Thus, the center of the spere is (-4, 3, -1) and the radius of the sphere is 3.

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