## What is a sphere ?

If 'C' is a fixed point in **space** and 'a' is a non-negative real number, then the set of all points 'P' in space, such that the distance CP = a, is called a **sphere**.

The point 'C' is it's center and 'a' it's radius.

A sphere with a = 0 (i.e, zero radius) is called **point sphere**.

** Another definition :** The locus of a variable point in space which moves such that its distance from a fixed point C in space is constant 'a' (≥ 0) is called a sphere.

We shall first discuss Vector and Cartesian equations of a sphere.

The concept of scalar product in vector algebra is used to derive the vector equation.

We shall use bold letters to represent a vector.

## Vector form of equation of a sphere

The vector equation of the sphere with center 'C', whose position vector is **c**, and radius 'a' is given by

| **r** – **c** | = a

or **r**^{2} – 2**r.****c** + **c**^{2} = a^{2}

Conversely, if P(**r**) is any point satisfying

| **r** – **c** | = a, then

CP = a and P lies on the sphere.

If OC = a (i.e, the origin of reference lies on the sphere), the equation of the sphere is

**r**^{2} – 2**r.****c** = 0

And if **c = 0** (i.e, the origin of reference is the center of the sphere), the equation of the sphere is

**r**^{2} = a^{2} or | **r** | = a

If A(**a**) and B(**b**) are the two end points of a diameter of the sphere, then its vector equation is

(**r** – **a**).(**r** – **b**) = 0

or **r**^{2} – **r**.(**a** + **b**) + **a**.**b** = 0

Let the center of the sphere be C(x_{1}, y_{1}, z_{1}). Let it's radius be 'a'.

If P(x, y, z) is a point on the sphere,

| CP | = a

or | CP |^{2} = a^{2}

⇒ ** (x – x _{1})^{2} + (y – y_{1})^{2} + (z – z_{1})^{2} = a^{2}**

which is the Cartesian equation of a sphere.

It is more common to denote the radius of a sphere by 'r' (and not 'a').

We now use the symbol **Σ** to represent a sphere with the center 'C' and radius 'r'.

Then in set-builder form

Σ = { P ∈ **R ^{3}** : CP = r}

If r = 0, Σ is called a point sphere.

The equation of a sphere whose center is A(x

_{0}, y

_{0}, z

_{0}) and radius 'r' is

**(x – x**

_{0})^{2}+ (y – y_{0})^{2}+ (z – z_{0})^{2}= r^{2}If the center is the origin (0, 0, 0), the equation is reduced to

**x**

^{2}+ y^{2}+ z^{2}= r^{2}## Diameter of a sphere :

If A and B are two points on Σ and if the line passes through its center, then is called as **diameter line**. The line sequent AB is called diameter of Σ.

The equation of a sphere on the join of A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) as diameter is

** (x – x _{1})(x – x_{2}) + (y – y_{1})(y – y_{2}) + (z – z_{1})(z – z_{2}) = 0 **

A diameter divides a sphere into two equal parts. Each is called a hemisphere.

If P(x_{0}, y_{0}, z_{0}) is a point on the sphere and is distinct from A and B, then

(x_{0} – x_{1})(x_{0} – x_{2}) + (y_{0} – y_{1})(y_{0} – y_{2}) + (z_{0} – z_{1})(z_{0} – z_{2}) = 0

We know (x_{0} – x_{1}, y_{0} – y_{1}, z_{0} – z_{1}) and (x_{0} – x_{2}, y_{0} – y_{2}, z_{0} – z_{2})
are **triads** of **direction numbers** and respectively.

This implies is perpendicular to .

Therefore the angle in a hemisphere is right angle(90°).

The equation of a sphere in **R ^{3}** is expressed of the form

x

^{2}+ y

^{2}+ z

^{2}+ 2ux + 2vy + 2wz + d = 0

where u

^{2}+ v

^{2}+ w

^{2}≥ d

The above form is called the general form of the equation of a sphere.

The center of above sphere is

**(– u, – v, – w)**

It's radius is given by

**√(u**

^{2}+ v^{2}+ w^{2}– d)
In general, the locus of a second degree equation in the three variables x, y, and z is

ax^{2} + by^{2} + cz^{2} + 2fyz + 2gzx + 2hxy + 2ux + 2vy + 2wz + d = 0

The locus becomes a sphere if

i) a = b = c ≠ 0

ii) f = g = h = 0

iii) u^{2} + v^{2} + w^{2} ≥ ad