      Custom Search   OTHER PARAMETRIC EQUATIONS EXAMPLE. Find the equations of the tangent line and the normal line and the lengths of the tangent and the normal for the curve represented by and at given that and SOLUTION.- Since t equals 1 we write x=1 and y=3 and so that The equation of the tangent line when t is equal to 1 is y-3=1(x-1) y=x+2 The equation of the normal line is y- 3 = -1(x- 1) y= -x+4 The length of the tangent is The length of the normal is EXAMPLE: Find the equations of the tangent line and the normal line and the lengths of the tangent and the normal to the curve represented by the parametric equations and at the point where given that and SOLUTION. We know that Then at the point where , we have If is substituted in the parametric equations, then and The equation of the tangent line when is or The equation of the normal is or x-y=0 The length of the tangent is The length of the normal is The horizontal and vertical tangents of a curve can be found very easily when the curve is represented by parametric equations. The slope of a curve at any point equals zero when the tangent line is parallel to the X axis. In parametric equations, if x = x(t) and y = y(t), then the horizontal and vertical tangents can be found easily by setting and For the horizontal tangent solve equals zero for t and for the vertical tangent solve equals zero for t. EXAMPLE: Find the points of contact of the horizontal and the vertical tangents to the curve represented by the parametric equations and Plot the graph of the curve by taking from 0° to 360° in increments of 30°, given that and  Figure 3-7.-Ellipse. SOLUTION: The graph of the curve shows that the figure is an ellipse, figure 3-7; consequently, it will have two horizontal and two vertical tangents. The coordinates of the horizontal tangent points are found by first setting This gives so that and Substituting 0°, we have and Substituting 180°, we obtain and The coordinates of the points of contact of the horizontal tangents to the ellipse are (3,1) and (3,7). The coordinates of the vertical tangent points of contact are found by setting We find from which Substituting 90°, we obtain and Substituting 270° gives and The coordinates of the points of contact of the vertical tangents to the ellipse are (- 1,4) and (7,4).