Answer:
The equation is [tex]x^2 + y^2 +z^2 + 14 x + 3y \ + 14z + 86.25=0[/tex]
Step-by-step explanation:
From the question we are told that
The diameter endpoints is (-8, -3, -10) and (-6, 1, -4)
Generally the equation of a sphere with center coordinates (a, b , c ) and radius r is mathematically represented as
[tex](x - a )^2 + (y -b )^2 + (z -c)^2 = r^2[/tex]
Now since we are given the endpoints of the diameter then we can obtain the center coordinates as follows
[tex](a, b , c) = [ \frac{ -8 +(-6)}{2} , \frac{-3 + (1)}{ 2} , \frac{ -10 + (-4)}{2} ][/tex]
[tex](a, b , c) = [ -7 , -1.5 , -7 ][/tex]
Now the length of the diameter is evaluated as
[tex]|d| = \sqrt{ (-8 - (-6 ))^2 + ( -3 - (1) )^2 + ( -10 - (-4))^2 }[/tex]
[tex]|d| = \sqrt{56 }[/tex]
[tex]|d| = \sqrt{4 * 14 }[/tex]
[tex]|d| = 2 \sqrt{ 14 }[/tex]
Now the radius is mathematically represented as
[tex]r = \frac{|d|}{2}[/tex]
[tex]r = \frac{ 2 \sqrt{14} }{2}[/tex]
[tex]r = \sqrt{14}[/tex]
So
[tex](x - -7 )^2 + (y --1.5 )^2 + (z --7)^2 = ( \sqrt{14} )^2[/tex]
[tex](x +7 )^2 + (y +1.5 )^2 + (z +7)^2 = 14[/tex]
[tex]x^2 + 14 x + 49 + y^2 + 3y + 2.25 +z^2 14z + 49 = 14[/tex]
[tex]x^2 + y^2 +z^2 + 14 x + 3y \ + 14z + 86.25=0[/tex]
