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.
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Solution
Coordinates of end points of diameter of sphere are (1,0,1) and (5,4,5) Equation of sphere (x−x1)(x−x2)+(y−y1)(y−y2)+(z−z2)(z−z2)=0 (x−1)(x−5)+(y−0)(y−4)+(z−1)(z−5)=0 x2−5x−x+5+y2−4y+z2−5z−z+5=0 x2+y2+z2−6x−4y−6z+10=0 this is the equation to the required sphere Coordinate of centre of sphere ∵ Midpoint of diameter AB is O and centre of sphere is O ∴ Coordinate of centre (x=5+12,y=4+02,z=5+12) (x=62,y=42,z=62) (3,2,3) Diameter of sphere = distance between points A and B =√(5−1)2+(4−0)2+(5−1)2=√16×3=4√3