Mathematics

Mathematics

Problem

Find the equation of the sphere whose centre is (3,−2,5)(3,-2,5) and radius is 4 units4\,\text{units}. a) (x−3)2+(y−2)2+(z−5)2=16(x-3)^2+(y-2)^2+(z-5)^2=16 b) (x−3)2+(y+2)2+(z−5)2=16(x-3)^2+(y+2)^2+(z-5)^2=16 c) (x−3)2+(y+2)2+(z−5)2=4(x-3)^2+(y+2)^2+(z-5)^2=4 d) (x−3)2+(y+2)2+(z−5)2=32(x-3)^2+(y+2)^2+(z-5)^2=32
Intuitive explanation

The equation of a sphere is really just the 3D version of a circle's equation—it captures all points at a fixed distance (the radius) from a center. You'll use the distance formula in three dimensions: the difference between any point (x,y,z)(x, y, z) on the sphere and the center must equal the radius. Square both sides to eliminate the square root and you'll get a clean, standard form.

Intuitive explanation

The equation of a sphere is really just the 3D version of a circle's equation—it captures all points at a fixed distance (the radius) from a center. You'll use the distance formula in three dimensions: the difference between any point (x,y,z)(x, y, z) on the sphere and the center must equal the radius. Square both sides to eliminate the square root and you'll get a clean, standard form.

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