1.

Obtain the equation of the sphere through the four points (4,-1,2); (0,-2,3); (1,-5,-1) and (2,0, 1).

Answer» Let the `center(a,b,c)` and `radius=r`
So, Equation of sphere be:`(x-a)^2+ (y-b)^2+ (z-c)^2=r^2`
It passes through the points `A(4.-1.2), B(0,-2,3), C(1,-5,-1) and D(2,0,1)`
By point A,`(4-a)^2+ (-1-b)^2+ (2-c)^2 =r^2`
By point B,`(0-a)^2+ (-2-b)^2 + (3-c)^2=r^2`
By point C,`(1-a)^2 + (-5-b)^2+ (-1-c)^2=r^2`
By point D,`(2-a)^2+ (0-b)^2+ (1-c)^2=r^2`
On solving We get,`a=2, b=-3, c=1, r=3`
Hence, Equation of sphere is=`(x-2)^2+ (y+3)^2 + (z-1)^2=9`


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