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Question

Find the equation of a sphere, whose centre is (1,1,1) and radius is 5 units


A

x2+y2+z2xyz=5

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B

x2+y2+z2xx2y2z=5

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C

x2+y2+z2xyz=25

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D

x2+y2+z22x2y2z=22

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Solution

The correct option is D

x2+y2+z22x2y2z=22


Sphere can be thought of as a three dimensional extension of a circle. Similar to distance formula and section formula, the equation of sphere has additional terms involving z coordinate. To find the equation of the sphere, let’s consider a general point P(x,y,z) on the sphere. Let the centre of the sphere be O(1,1,1). We are given that the radius is 5 units.

Distance of P from O will be equal to the radius, that is 5 units. Using distance formula, we get

PO=(x1)2+(y1)2+(z1)2=5(x1)2+(y1)2+(z1)2=25x2+y2+z22x2y2z=22
This is the equation of the sphere with centre (1,1,1) and radius 5 units


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