Question

To find: An equation of the largest sphere when it is contained in the first octant with center\((5,4,9)\).

Short answer

The equation of the largest sphere that contained in the first octant is \({(x - 5)^2} + {(y - 4)^2} + {(z - 9)^2} = 16\)

Step-by-step solution

Formula used to express the equation of a sphere

Write the expression to find an equation of a sphere with centre\(C(h,k,l)\)and radius\(r\).

\({(x - h)^2} + {(y - k)^2} + {(z - l)^2} = {r^2}\) …… (1)

Here,

\((h,k,l)\)is the centre of a sphere and

\(r\)is the radius of a sphere.

Find the radius of largest sphere when it is contained in the first octant with center\((5,4,9)\)

The radius of the largest sphere that contained in the first octant is the minimum distance from the center to the any of the three coordinate planes.

The distance from any point to the\(xy\)plane is the absolute value of\(z\)-coordinate of the point.

The absolute value of\(z\)coordinate of the point\((5,4,9)\)is\(\left| 9 \right|\), which is 9 . Thus, the distance from the center to the\(xy\)-plane is 9 .

The distance from any point to the\(yz\)-plane is the absolute value of\(x\)coordinate of the point.

The absolute value of\(x\)coordinate of the point\((5,4,9)\)is\(\left| 5 \right|\), which is\(5\). Thus, the distance from the center to the\(yz\)plane is\(5\).

The distance from any point to the\(xz\)-plane is the absolute value of\(y\)coordinate of the point.

The absolute value of\(y\)-coordinate of the point\((5,4,9)\)is\(\left| 4 \right|\), which is\(4\). Thus, the distance from the center to the\(xz\)-plane is 4 .

Among the calculated three distances, the distance from the center of the sphere to the\(xz\)plane is the minimum value.

Therefore, the radius of the largest sphere that contained in the first octant is \(4\).

Determine the equation of largest sphere when it is contained in the first octant with center\((5,4,9)\)

Calculation of equation of the sphere:

In equation (1), substitute\(5\)for\(h\),\(4\)for\(k\),\(9\)for\(l\), and\(4\)for\(r\).\(\begin{aligned}{l}{(x - 5)^2} + {(y - 4)^2} + {(z - 9)^2} = {(4)^2}\\{(x - 5)^2} + {(y - 4)^2} + {(z - 9)^2} = 16\end{aligned}\)

Thus, the equation of the largest sphere that contained in the first octant is \({(x - 5)^2} + {(y - 4)^2} + {(z - 9)^2} = 16\).