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Q)

Ho to find the equations of a sphere which passes through three given point

Ho to find the equations of a sphere which passes through three given points and touches a given plane

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A)
Let the radius of the sphere be r.

The sphere touches the three coordinate planes.

⇒ The centre of the sphere is equidistant from all the three coordinate planes and this distance is also the radius of the sphere.

⇒ The coordinates of the centre of the sphere is (r,r,r).

⇒ The equation of the sphere touching the three coordinate planes is

(x−r)^2+(y−r)^2+(z−r)^2=r^2

⇒x^2+y^2+z^2−2r(x+y+z)+2r^2=0
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