Answer
Every normal line to the sphere passes through the center of the sphere.
Work Step by Step
Our aim is to determine the normal line equation.
The general form is:
$\dfrac{(x_2-x_1)}{f_x(x_1,y_1,z_1)}=\dfrac{(y_2-y_1)}{f_y(x_1,y_1,z_1)}=\dfrac{(z_2-z_1)}{f_x(x_1,y_1,z_1)}$ ...(1)
From the given data, we have $f(x,y,z)=(p,q,r)$
Equation (1), becomes:
Thus,
$\dfrac{(x-p)}{2p}=\dfrac{(y-q)}{2q}=\dfrac{(z-r)}{2r}$
After simplifications, we get:$\dfrac{x}{p}=\dfrac{y}{q}=\dfrac{z}{r}$
Hence, $x=y=z=0$
This has been proved that every normal line to the sphere passes through the center of the sphere.
