Question

The coordinates of ends of one diameter of a sphere are $(1,0,1)$ and $(5,4,5)$. Find the equation, centre and diameter of sphere.

Answer 1

Coordinates of end points of diameter of sphere are $(1,0,1)$ and $(5,4,5)$
Equation of sphere
$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)+(z-z_2)(z-z_2)=0$
$(x-1)(x-5)+(y-0)(y-4)+(z-1)(z-5)=0$
$x^2-5x-x+5+y^2-4y+z^2-5z-z+5=0$
$x^2+y^2+z^2-6x-4y-6z+10=0$
this is the equation to the required sphere
Coordinate of centre of sphere
$\because$ Midpoint of diameter $AB$ is $O$ and centre of sphere is $O$
$\therefore$ Coordinate of centre
$(x=\frac{5+1}{2},y=\frac{4+0}{2},z=\frac{5+1}{2})$
$(x=\frac{6}{2},y=\frac{4}{2},z=\frac{6}{2})$
$(3,2,3)$
Diameter of sphere $=$ distance between points $A$ and $B$
$=\sqrt{{(5-1)}^2+{(4-0)}^2+{(5-1)}^2}=\sqrt{16\times 3}=4\sqrt{3}$