In the given figure, O and O' are centers of two circles, touching each other at point P. The common tangent at P meets a direct common tangent AB at M. Show that :

(i) M is mid-point of AB.

(ii) ∠APB = 90°

AB is a line segment joining the points A(3, 2) and B(4, 1). Find :

(i) the ratio in which AB is divided by point P on x-axis

(ii) the co-ordinates of point P

(iii) the equation of the line that passes through the point P and is perpendicular to AB.

An express train makes a run of 240 km at a certain speed. Another train whose speed is 12 km/h less takes an hour longer to cover the same distance. Find the speed of the express train.

Draw and describe the locus in each of the following cases :

(i) The locus of vertices of all isosceles triangles having a common base.

(ii) The locus of points inside a circle and equidistant from two fixed points on the circle.

(iii) The locus of centres of all circles passing through two fixed points.

(iv) The locus of a point in rhombus ABCD which is equidistant from AB and AD.