Question

How do I find the equation of a sphere that passes through the origin and whose center is $\left( {4,1,2} \right)$?

Answer 1

In order to solve this question, we use the formula for the general equation of a sphere. Then, we need to find the value of the radius $r$ , hence we have to calculate the radius, so that we use the second point given – the points for the origin, Finally we find the value of the radius and place it in the equation along with the coordinates of the Centre of the sphere.

Formula used: ${\left( {x - {x_c}} \right)^2} + {\left( {y - {y_c}} \right)^2} + {\left( {z - {z_c}} \right)^2} = {r^2}$
${\left( {{x_c} - {x_o}} \right)^2} + {\left( {{y_c} - {y_o}} \right)^2} + {\left( {{z_c} - {z_o}} \right)^2} = {r^2}$

Complete step by step solution:
In this question, we are asked to find the equation of a sphere which passes through the origin and has its center at $\left( {4,1,2} \right)$.
As we know that the general formula for finding the equation of a sphere is given as:
${\left( {x - {x_c}} \right)^2} + {\left( {y - {y_c}} \right)^2} + {\left( {z - {z_c}} \right)^2} = {r^2}$
Here the coordinates $\left( {{x_c},{y_c},{z_c}} \right)$ refer to the coordinates of the centre C of the sphere while $r$ refers to the radius of the sphere.
Now we know that the sphere passes through the origin, which simply means that the sphere passes through the coordinates $\left( {{x_o},{y_o},{z_o}} \right) = \left( {0,0,0} \right)$
To calculate the radius, we use the second point given – the points for the origin.
As radius simply means the distance of the centre from the point through which the sphere passes.
Therefore, ${\left( {{x_c} - {x_o}} \right)^2} + {\left( {{y_c} - {y_o}} \right)^2} + {\left( {{z_c} - {z_o}} \right)^2} = {r^2}$
$\Rightarrow {\left( {4 - 0} \right)^2} + {\left( {1 - 0} \right)^2} + {\left( {2 - 0} \right)^2} = {r^2}$
Simplifying it further, we get:
$\Rightarrow 16 + 1 + 4 = {r^2}$
On adding the terms, we get:
$\Rightarrow 21 = {r^2}$
Therefore, the equation of a sphere is given as ${\left( {x - {x_c}} \right)^2} + {\left( {y - {y_c}} \right)^2} + {\left( {z - {z_c}} \right)^2} = {r^2}$
Substituting the values:
$\Rightarrow {\left( {x - 4} \right)^2} + {\left( {y - 1} \right)^2} + {\left( {z - 2} \right)^2} = 21$

The equation of sphere passing through $(4,1,2)$ is ${\left( {x - 4} \right)^2} + {\left( {y - 1} \right)^2} + {\left( {z - 2} \right)^2} = 21$.

Note: A sphere is a three dimensional figure. The half of a sphere is known as the hemisphere. If we compare a sphere to a circle, we find that the circle is a two-dimensional figure, while the sphere is a three dimensional object. Three dimensional figures make use of the three axes, which are x,y and z while two dimensional figures only make use of the first two axes.