Find the equation of the perpendicular bisector of the line joining the points \((x_1, y_1)\), \((x_2, y_2)\). A fixed circle has centre \(C\) and radius \(2a\). \(A\) is a fixed point inside the circle and \(P\) is a variable point on the circumference. Prove that the perpendicular bisector of \(AP\) touches the foci are at \(C\) and \(A\), and whose major axis is of length \(2a\).
Two planes \begin{align} x - 3y + 2z &= 2, \\ 2x - y - z &= 9, \end{align} meet in the line \(l\). Find the equations of (i) the plane through the origin which contains \(l\), (ii) the plane through the origin which is perpendicular to \(l\). Find also the coordinates of the reflection of the origin in \(l\).
The surfaces of two spheres have more than one real common point. Prove that they intersect in a circle. A triangle \(BCD\) is given in a plane \(\alpha\). Prove that there are just two possible positions, reflections of each other in \(\alpha\), for a point \(A\), which is such that the angles \(BAC\), \(CAD\), \(DAB\) are right angles, if and only if the triangle \(BCD\) is acute-angled. Find the distance of \(A\) from \(\alpha\) in the case where \(BCD\) is an equilateral triangle with sides of unit length.