Question: (4 points) In what follows, notice that the tangent plane to a given point of a sphere is the plane passing through the point and perpendicular to the line connecting it with the centre of the sphere; also, two spheres are said to be tangent if they have a common point where the tangent planes coincide.

1. Let \(S_{1}\) and \(S_{2}\) be two non-concentric spheres given by:
   \[\begin{aligned} &f_{1}(\mathrm{r})=0 \\ &f_{2}(\mathrm{r})=0 \end{aligned}\]
   where \(\mathbf{r}=(x, y, z)\) is the position vector of each point in space, and \(f_{1}, f_{2}\) are quadratic polynomials in $x, y, z$ such that the coefficient of \(x^{2}\) is 1. Show that the equation \(f_{1}(\mathrm{r})-f_{2}(\mathrm{r})=0\) defines a plane. Show also that if the spheres are tangent this plane coincide with their common tangent plane.
2. Consider the spheres:

Are the spheres tangent? Find an equation for their common tangent plane.

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