Answer to Question #297072 in Analytic Geometry for Rio

Question

<e> Find the equation of the sphere touching the plane 8𝑥 + 5𝑦 + 3𝑧 + 1 = 0 at (3, −1, −1) and cutting the sphere 𝑥² + 𝑦² + 𝑧² − 2𝑥 + 𝑦 − 𝑧 − 6 = 0 orthogonally.

Answer 1

At first, we check that the point $(3,-1,-1)$ belongs to the plane: $3\cdot8-5-3 +1=17$. The point does not belong to the plane. However, we present the general scheme for derivation of equation of the sphere. An orthogonal cut of sphere given in the formulation of the task can be received from a plane that passes through the center of the sphere. This sphere can be rewritten as: $(x-1)^2+(y+\frac{1}{2})^2+(z-\frac{1}{2})^2-7.5=0$. The center is: $x=1,y=-\frac{1}{2},z=\frac{1}{2}$. The general equation of the plane is: $ax+by+cz+d=0$. Coefficients have to satisfy relation: $a-\frac{1}{2}b+\frac{1}{2}c+d=0$. From the latter we receive the equation: $(\frac{1}{2}b-\frac{1}{2}c-d)x+by+cz+d=0$. Thus, to get the equation of the sphere we take the equation in the form: $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2$ and solve the system:

$\left\{\begin{array}{llll}(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2\\(x-1)^2+(y+\frac{1}{2})^2+(z-\frac{1}{2})^2-7.5=0\\(\frac{1}{2}b-\frac{1}{2}c-d)x+by+cz+d=0\\x=x_1,y=y_1,z=z_1\end{array}\right.$

$x_1,y_1,z_1$ are coordinates of the point that belongs to the plane $8x+5y+3z+1=0$