In equation (1), substitute \(2{\rm{ for }}h, - 6{\rm{ for }}k,4{\rm{ for }}l{\rm{, and }}5{\rm{ for }}r{\rm{. }}\)
\(\begin{aligned}{l}{(x - 2)^2} + {(y - ( - 6))^2} + {(z - 4)^2} = {5^2}\\{(x - 2)^2} + {(y + 6)^2} + {(z - 4)^2} = 25\end{aligned}\)
Thus, the equation of the sphere with the center\((2, - 6,4)\) and radius\(5\)is\({(x - 2)^2} + {(y + 6)^2} + {(z - 4)^2} = 25\).
The intersection of the sphere with the\(xy\)- plane is the set of all points on the sphere whose\(z\)- coordinates are zero.
Therefore, the intersection of the sphere with the\(xy\)- plane is obtained by substituting 0 for\(z\) in the equation of the sphere
The equation of the sphere is also written as follows.
\({(x - 2)^2} + {(y + 6)^2} + {(z - 4)^2} = {5^2}\) …… (2)
In equation (2), substitute\(0{\rm{ for }}z\)to find the intersection of the sphere with the\(xy\)- plane.
\(\begin{aligned}{l}{(x - 2)^2} + {(y + 6)^2} + {(0 - 4)^2} = {5^2}\\{(x - 2)^2} + {(y + 6)^2} + 16 = 25\\{(x - 2)^2} + {(y + 6)^2} = 9\\{(x - 2)^2} + {(y - ( - 6))^2} = {3^2}\end{aligned}\) …… (3)
The equation (3) represents the circle in the\(xy\)- plane with center\({\rm{(2, - 6,0)}}\) and radius\({\rm{3}}{\rm{.}}\)
Thus, the intersection of the sphere with the\(xy\)- plane is a circle in the\(xy\)- plane with center\({\rm{(2, - 6,0)}}\)and radius\({\rm{3}}{\rm{.}}\)
The intersection of the sphere with the\({\rm{yz}}\)- plane is the set of all points on the sphere whose\({\rm{x}}\)- coordinates are zero.
Therefore, the intersection of the sphere with the\({\rm{yz}}\)- plane is obtained by substituting\({\rm{0}}\) for\({\rm{x}}\) in the equation of the sphere.
In equation (2), substitute\({\rm{0 for x}}\) to find the intersection of the sphere with the\({\rm{yz}}\)- plane.
\(\begin{aligned}{l}{(0 - 2)^2} + {(y + 6)^2} + {(z - 4)^2} = {5^2}\\4 + {(y + 6)^2} + {(z - 4)^2} = 25\\{(y + 6)^2} + {(z - 4)^2} = 21\\{(y - ( - 6))^2} + {(z - 4)^2} = {(\sqrt {21} )^2}\end{aligned}\) …… (4)
The equation (4) represents a circle in the\({\rm{yz}}\)- plane with center\((0, - 6,4)\) and radius\(\sqrt {21} \).
Thus, the intersection of the sphere with the\({\rm{yz}}\)- plane is a circle in the\({\rm{yz}}\)- plane with center\((0, - 6,4)\)and radius\(\sqrt {21} \).
The intersection of the sphere with the $x z$-plane is the set of all points on the sphere whose\(y\) - coordinates is zero.
Therefore, the intersection of the sphere with the\(xz\)- plane is obtained by substituting\(0\)for\(y\) in the equation of the sphere.
In equation (2), substitute\(0\) for\(y\) to find the intersection of the sphere with the\(xz\)- plane.
\(\begin{aligned}{l}{(x - 2)^2} + {(0 + 6)^2} + {(z - 4)^2} = {5^2}\\{(x - 2)^2} + 36 + {(z - 4)^2} = 25\\{(x - 2)^2} + {(z - 4)^2} = - 11\end{aligned}\) …… (5)
The equation (5) does not satisfy any point on the sphere.
The absolute value of\( - 11\) which is\(11\) , indicates the distance from the center of the sphere to the\(xz\)- plane. It is also greater than the radius of the sphere (5).
Therefore, the sphere does not intersect with the\(xz\)- plane
