Consider the sphere of radius \(a\) and centre the origin.
%Show that the line through the point with position vector
%\({\bf b}\) and parallel to a unit
%vector \({\bf m}\) intersects the sphere at two points if
%$$
%a^2 > {\bf b}.{\bf b} -({\bf b}.{\bf m})^2 \,.
%$$
%What is the corresponding condition for there to be precisely one
%point of intersection?
%If this point has position vector \({\bf p}\), show that the line
%is perpendicular to \({\bf p}\).
Show that the line \({\bf r} ={\bf b} + \lambda {\bf m}\),
where \(\bf m\) is a unit vector,
intersects the sphere \({\bf r}\cdot {\bf r} = a^2\) at two points if
$$
a^2 > {\bf b}\cdot{\bf b} -({\bf b}\cdot{\bf m})^2 \,.
$$
Write down the corresponding condition for there to be precisely one
point of intersection.
If this point has position vector \({\bf p}\), show that \({\bf m}\cdot{\bf p}=0\).
Now consider a second sphere of radius \(a\)
and a plane perpendicular to a unit vector~\({\bf n}\).
The centre of the sphere
has position vector \({\bf d}\)
and the
minimum distance from the origin to the plane is \(l\). What is the
condition for the plane to be tangential to this second sphere?
Show that the first and second spheres intersect at right angles
({\em i.e.\ }the two radii to each point of
intersection are perpendicular) if
$$
{\bf d}\cdot{\bf d} = 2 a^2 \,.
$$