The sphere

Problem

A sphere is centered at \(M(-3|2|7)\). The point \(P(3|4|4)\) is located on the surface of this sphere.

1. The point \(Q\) is also located on the surface of the sphere. The line segment \([PQ]\) contains the center \(M\). Determine the coordinates of the point \(Q\).

2. Prove that the sphere touches the \(x_1x_2\)-plane.

Solution of part a

Since both points, \(P\) and \(Q\), are located on the surface of the sphere and the line segment connecting these points contains the center of the sphere, \(Q\) can be represented as:

\[\vec{Q} = \vec{M} + \left(\vec{M}-\vec{P}\right).\]

As a result, we obtain

\[Q(-9|0|10),\]

which can be verified with Sage:

Solution of part b

In order to verify whether the sphere touches the \(x_1x_2\)-plane, we only need to calculate the distance between the plane and the center of the sphere and compare it to the sphere’s radius. This distance is equal to the absolute value of the center’s \(x_3\)-coordinate, which equals 7.

The radius of the sphere can be determined from the distance between the point \(P\) on the sphere and the sphere’s center:

\[r = \left\vert\vec{M}-\vec{P}\right\vert = \sqrt{(-3-3)^2 + (2-4)^2 + (7-4)^2} = 7\]

Thus, the distance between the plane and the center of the sphere equals the radius of the sphere, so that the sphere touches the plane in one point. The coordinates of this point can be obtained by projecting the sphere’s center to the \(x_1x_2\)-plane, yielding \(S(-3|2|0)\).

We can use Sage to illustrate this in three-dimensional space. The mouse can be used to rotate the view.