<s>[INST] How do I can simplify the below intersection? let $S$ be a sphere defined by this equation :$ \left(x-\frac{4}{3}\right)^2+\left(y-\frac{1}{3}\right)^2+\left(z-\frac{5}{3}\right)^2=\frac{25}{36}$ and $(p)$ the plane defined by the following equation :$2x+y-2z+4=0$ , 


My question here is : How do I can get the intersection between $(p)$ and $(S)$?  By the way I want to get the center and the radius of the circle which it is the geometrical intersection of $(p)$ and $(S)$ .


Attempt : I took $y=2(z-x-2)$ from the equation of $(p)$ and by substitution in $(S)$ I have got this complicated form 
 [/INST] \begin{align*}
  \text{Radius of the sphere} &= \frac{5}{6} \\
  \text{Distance of the centre from the plane} &=
  \frac{2\left( \frac{4}{3} \right)+
         \left( \frac{1}{3} \right)-
        2\left( \frac{5}{3} \right)+4}
       {\sqrt{2^2+1^2+2^2}} \\
  &= \frac{11}{3} \\
  &> \frac{5}{6}
\end{align*}

Hence, no intersections between plan and sphere.

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