How to write the equation of the sphere (2 cases)

#2. When you know the four points that pass through

The sphere equation $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2$ can also be written as an expansion $x^2+y^2 +z^2-2x_0x-2y_0y-2z_0z+d=0$ with $d=x_0^2+y_0^2+z_0^2-R^2$

When asked to write the equation of a sphere passing through four points we will use the equation in expanded form

Suppose the required sphere equation passes through four points $A=(x_a; y_a; z_a), B=(x_b; y_b; z_b), C=(x_c; y_c; z_c), D=(x_d; y_d; z_d)$

Step 1. Substituting the coordinates of the four points A, B, C, D in the expansion equation, respectively, we will get a system of four quadratic equations with four unknowns:

$\left\{\begin{array}{ll}x_a^2+y_a^2+z_a^2-2x_0x_a-2y_0y_a-2z_0z_a+d&=0\\x_b^2+y_b^2+z_b^2-2x_0x_b- 2y_0y_b-2z_0z_b+d&=0\\x_c^2+y_c^2+z_c^2-2x_0x_c-2y_0y_c-2z_0z_c+d&=0\\x_d^2+y_d^2+z_d^2-2x_0x_d-2y_0z_d-2+z d&=0\end{array}\right.$

Step 2. Use method substitution methodthe addition method, … to solve this system we will find the solution $x_0, y_0, z_0, d$

Step 3. Substituting $x_0, y_0, z_0, d$ into the equation $x^2+y^2+z^2-2x_0x-2y_0y-2z_0z+d=0$ we get a sphere equation

Attention:

• Phrases four points on the sphere, four points on the sphere, the sphere passes through four points is the same meaning
• There are infinitely many equations of the sphere passing through the three points

Example 3. Write the equation of the sphere passing through the four points $A(2; 4; -1), B(1; 4; -1), C(2; 3; 4), D(2; 2; -1)$

The answer:

The required sphere equation has the form $x^2+y^2+z^2-2x_0x-2y_0y-2z_0z+d=0$ and passes through four points $A(2; 4; -1), B(1 ; 4; -1), C(2; 3; 4), D(2; 2; -1)$ so we get the system of equations:

$\left\{\begin{array}{ll}2^2+4^2+(-1)^2-2x_02-2y_04-2z_0(-1)+d&=0\\1^2+4^2 +(-1)^2-2x_01-2y_04-2z_0(-1)+d&=0\\2^2+3^2+4^2-2x_02-2y_03-2z_04+d&=0\\2^2+ 2^2+(-1)^2-2x_02-2y_02-2z_0(-1)+d&=0\end{array}\right. \Leftrightarrow \left\{\begin{array}{ll}-4x_0-8y_0+2z_0+d&=-21\\-2x_0-8y_0+2z_0+d&=-18\\-4x_0-6y_0-8z_0+d&=- 29\\-4x_0-4y_0+2z_0+d&=-9\end{array}\right. \Leftrightarrow \left\{\begin{array}{ll}x_0&=\frac{3}{2}\\y_0&=3\\z_0&=\frac{7}{5}\\d&=\frac{31} {5}\end{array}\right.$

So => the equation for the demand surface is $x^2+y^2+z^2-3x-6y-\frac{14}{5}z+\frac{31}{5}=0$

#3. Quick trick to write the equation of a sphere passing through four points using a CASIO . calculator

The foundation of this trick is the equation $x^2+y^2+z^2-2x_0x-2y_0y-2z_0z+d=0$ and the calculation method Equation / Func