Equation of Sphere - Practice Questions & MCQ

A Sphere is the locus of a point which moves in space in such a way that its distance from a fixed point always remains constant. The fixed point is called the centre of the sphere and the fixed distance is called the radius of the sphere.

Equation of Sphere

Sphere's are the 3D representations of circles. The equation of a sphere is similar to that of a circle but with an extra variable for the extra dimension.

Let C(a, b, c) be the centre of the sphere with radius r. Let P(x, y, z) be any point on the sphere,

Then,

$
\begin{aligned}
C P & =r \\
|C P| & =r
\end{aligned}
$

From the distance formula
$
\begin{array}{cc}
\Rightarrow & C P^2=r^2 \\
\Rightarrow & (x-a)^2+(y-b)^2+(z-c)^2=r^2
\end{array}
$
since, $P(x, y, z)$ is an arbitrary point on the sphere, therefore required equation of the sphere is
$
(x-a)^2+(y-b)^2+(z-c)^2=r^2
$

NOTE:
If the centre of the sphere at the origin the equation of the sphere is $x^2+y^2+z^2=r^2$.
The equation of the sphere can also be written as $x^2+y^2+z^2-2(a x+b y+c z)+d=0$, where $d=a^2+b^2+c^2-r^2$