12.1. INTRODUCTION

In this chapter, we will discuss such geometrical and physical problems which lead to the differential equations of the first order and first degree.

A summary of the fundamental principles required in the formation of such differential equations is given in each case.

12.2. GEOMETRICAL APPLICATIONS

1. Cartesian Coordinates. Let P( x, y) be any point on the curve AB whose Cartesian equation is f( x, y) = 0. Let the tangent and the normal at P meet the x-axis in T and N respectively. Draw the perpendicular PM on the x-axis. Let ?MTP = ?, then

   1. Slope of tangent at

   2. Equation of the tangent at P is

      

      where (X, Y) are coordinates of any point on the line PT

   3. x-intercept of tangent = [by setting Y = 0]

      y-intercept of tangent = [by setting X = 0]

   4. Equation of the normal at P is

   5. Length of the tangent

   6. Length of the normal

   7. Length of the sub-tangent

   8. Length of the sub-normal

   9. If arc AP = s, then

1. Polar Coordinates. Let P( r, ?) be any point on the curve AB whose polar equation is r = f ( ?).

   Draw a line TON at right angles to the radius vector OP. Let the tangent and the normal...