STEP III 1998 question 8 solution

TODO: write up the question and add solutions for parts (ii) and (iii)

(i) TODO

(ii) By definition of a plane perpendicular to unit vector with minimum distance from the origin, the points in the plane satisfy

The points of the second sphere satisfy

For the the plane to be tangential to the second sphere, it must meet the radius vector (from the centre of the second sphere to the point of contact) at a right angle.

Let the point of contact be The radius vector is which has length and is parallel to

Since the scalar product of two parallel vectors is the product of their magnitudes:

Since, by definition, is in the plane:

Therefore:

This is the condition required.

(iii) TODO