How can we write the equation of a sphere in standard form?

A sphere is a three dimensional object. It's distance is same from any fixed point and is always a constant.

Answer: The equation of a sphere in standard form is x² + y² + z² = r². 

Let us see how is it derived.

Explanation:

Let A (a, b, c) be a fixed point in the space, r be a positive real number and P (x, y, z ) be a moving point such that AP = r is a constant.

⇒ AP = r

On squaring both the sides, we get

⇒ (AP)² = r² 

⇒ (x - a)² + ( y - b)² + ( z - c)² = r²  (By using distance formula)

This is the equation of a sphere with centre A (a, b, c) and radius r.

For equation of a sphere in standard form,

Let the centre be O (0, 0, 0) and P (x, y, z) be any point on the sphere as shown in the figure below,

Here, A (a, b, c) = O (0, 0, 0)

OP = r

⇒ OP² = r²

By using the distance formula, we get

⇒ (x - 0)² + (y - 0)² + (z - 0)² = r²   

⇒  x ² +  y² +  z² = r²

Thus, equation of a sphere in standard form is x² + y² + z² = r².