      Custom Search   EQUATIONS AND LENGTHS OF TANGENTS AND NORMALS In figure 3-5, the coordinates of point P1 on the curve are (x1,y1). Let the slope of the tangent line to the curve at point P1 be denoted by m1. If you know the slope and a point through which the tangent line passes, you can determine the equation of that tangent line by using the pointslope form. Thus, the equation of the tangent line, MP1, is The normal to a curve at a point (x1,y1) is the line perpendicular to the tangent line at that point. The slope of the normal line, m2, is  Figure 3-S.-Curve with tangent and normal lines. where, as before, the slope of the tangent line is m1. This is shown in the following: If then therefore, The equation of the normal line through P1 is Notice that since the slope of the tangent line is m, and the slope of the line which is normal to the tangent is m2 and then the product of the slopes of the tangent and normal lines equals -1. The relationship between the slopes of the tangent and normal lines is stated more formally as follows: The slope of the normal line is the negative reciprocal of the slope of the tangent line. Another approach to show the relationship between the slopes of the tangent and normal lines follows: The inclination of one line must be 900 greater than the other. Then if then We know from trigonometry that Therefore, The length of the tangent is defined as that portion of the tangent line between the point P1 (x1,y1) and the point where the tangent line crosses the X axis. In figure 3-5, the length of the tangent is the line MP,. The length of the normal is defined as that portion of the normal line between the point P1 and the X axis; that is the line, P1R1 which is perpendicular to the tangent line. As shown in figure 3-5,.the lengths of the tangent and normal may be found by using the Pythagorean theorem. From triangle MP1N, in figure 3-5, and Since then In triangle NP1R, and Thus, the length of the tangent is equal to and the length of the normal is equal to EXAMPLE. Find the equation of the tangent line, the equation of the normal line, and the lengths of the tangent and the normal of SOLUTION: Find the value of 2a from Since then The slope is Using the point-slope form of a straight line, we have then, at point (3,2) and which is the equation of the tangent line. Use the negative reciprocal of the slope to find the equation of the normal line as follows: then To find the length of the tangent, we use the Pythagorean theorem. Thus, the length of the tangent is The length of the normal is equal to PRACTICE PROBLEMS: Find the equations of the tangent line and the normal line, and the lengths of the tangent and the normal for the following:     Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business