CHAPTER IX 



80. The Equation of the Tangent to a Given Curve. The equation 

 of a line is y = mx + c where m is the tangent of the angle of 

 slope, or if the line is inclined to the axis of x at an angle 0, thsn 

 m = tan 0. 



The tangent to a curve is a straight line which passes through 

 a given point on the curve and also has the same slope as the 

 curve at that point. 



Let the co-ordinates of a point on a curve be h, k. Then the 



slope of the curve at that point is the value of -^- when x = h, 



dx 



and this must be the slope of the tangent. 



The equation of the tangent is y = x (~) + c. 

 But this line also passes through the point, 

 and 



, fdii\ 



Jc = h( -/-} + c 

 \dxj x=h 



_, , 

 Therefore 



The normal to a curve is the line drawn perpendicular to the 

 tangent through the point of contact. 



If the tangent to a curve is inclined to the axis of x at an angle 

 0, then the corresponding normal is inclined to the axis of x at 

 an angle (90 + 0). 



Slope of the normal = tan (90 + 0) 



-- cot 



= -- , where m = tan 

 m 



Then the slope of the normal to the curve at the given point is 



or 



130 



