# 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 x^{2 }+ y^{2 }+ z^{2 }= r^{2}.

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} = r^{2 }

⇒ (x - a)^{2} + ( y - b)^{2} + ( z - c)^{2 }= r^{2} (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

⇒ OP^{2} = r^{2}

By using the distance formula, we get

⇒ (x - 0)^{2} + (y - 0)^{2} + (z - 0)^{2 }= r^{2 }

⇒ x ^{2} + y^{2} + z^{2 }= r^{2}

### Thus, equation of a sphere in standard form is x^{2 }+ y^{2 }+ z^{2 }= r^{2}.

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