Question

Find the equation of the Sphere which passes through the points (0,0,2),(0,2,0) and (2,0,0) and whose centre lies on the plane \[x + y + z = 2\]

Answer 1

Hint: Recall the general equation of the sphere, the points given in the question will satisfy the equation of the sphere and try to get 3 separate equations using those 3 points. Solve those equations to get the desired values and at last put them in the general equation to get the final equation.
Complete Step by Step solution:
Let the equation of sphere.
\[{x^2} + {y^2} + {z^2} + 2ux + 2vy + 2wz + d = 0......................................................(i)\]
Where (-u,-v,-w) are the coordinates of the center of sphere
Because the points (0,0,2) lie on the sphere, hence.
\[\begin{array}{l}
\therefore {0^2} + {0^2} + {2^2} + 2u.0 + 2v.0 + 2w.2 + d = 0\\
 \Rightarrow 4w + d = - 4.......................................(ii)
\end{array}\]
Again (0,2,0) lies on sphere hence
\[\begin{array}{l}
\therefore {0^2} + {2^2} + {0^2} + 2u.0 + 2v.2 + 2w.0 + d = 0\\
 \Rightarrow 4v + d = - 4.......................................(iii)
\end{array}\]
And the last point (2,0,0), Putting it we will get as
\[\begin{array}{l}
\therefore {2^2} + {0^2} + {0^2} + 2u.2 + 2v.0 + 2w.0 + d = 0\\
 \Rightarrow 4u + d = - 4.......................................(iv)
\end{array}\]
Since centre of the sphere (−u,−v,−w) lies on the plane \[x + y + z = 2\]
\[\therefore - u - v - w = 2..............................(v)\]
Now by solving equations (ii) and (iii)
We will get
(ii)-(iii)
\[\begin{array}{l}
\therefore 4w + d - 4v - d = - 4 + 4\\
 \Rightarrow 4(w - v) = 0\\
 \Rightarrow w - v = 0\\
 \Rightarrow w = v
\end{array}\]
Similarly by solving (iii) and (iv)
(iii)-(iv)
\[\begin{array}{l}
\therefore 4v + d - 4u - d = - 4 + 4\\
 \Rightarrow 4(v - u) = 0\\
 \Rightarrow v - u = 0\\
 \Rightarrow v = u
\end{array}\]
From here we can say that \[v = u = w\]
Now replacing v and w with u in equation (v)
\[\begin{array}{l}
\therefore - u - v - w = 2\\
 \Rightarrow - u - u - u = 2\\
 \Rightarrow - 3u = 2\\
 \Rightarrow - u = \dfrac{2}{3}
\end{array}\]
Now as we know that \[v = u = w\]
Therefore, \[u = v = w = - \dfrac{2}{3}\]
Now for getting the value of d we will put the value of u in equation (iv)
\[\begin{array}{l}
\therefore 4u + d = - 4\\
 \Rightarrow 4 \times \dfrac{-2}{3} + d = - 4\\
 \Rightarrow \dfrac{-8}{3} + d = - 4\\
 \Rightarrow d = - 4 - \dfrac{8}{3}\\
 \Rightarrow d = \dfrac{{ - 12 + 8}}{3}\\
 \Rightarrow d = - \dfrac{{4}}{3}
\end{array}\]

Putting the value of u,v,w and d in the general equation of a sphere which is equation (i) we get the final equation as
\[\begin{array}{*{20}{l}}
{\therefore {x^2} + {y^2} + {z^2} + 2\left( { - \dfrac{2}{3}} \right)x + 2\left( { - \dfrac{2}{3}} \right)y + 2\left( { - \dfrac{2}{3}} \right)z - \dfrac{{4}}{3} = 0}\\
{ \Rightarrow {x^2} + {y^2} + {z^2} - \dfrac{4}{3}x - \dfrac{4}{3}y - \dfrac{4}{3}z - \dfrac{{4}}{3} = 0}\\
{ \Rightarrow 3{x^2} + 3{y^2} + 3{z^2} - 4x - 4y - 4z - 4 = 0}
\end{array}\]

Note: Solve the equations (ii),(iii),(iv) and (v) carefully the slightest error can lead to a different answer. By solving all these equations you will get the value of u,v and w only and then I have put in equation (iv) to get the value of d. You can also put it in equation (ii) or (iii) still you will get the same value.