THE TANGENT AND NORMAL TO A CURVE 181 



The equation to the normal is y = - x -\ -f c l 



\ . ' /i 



But this line passes through the given point, 



Jdx\ 



and k = h( -3- ) + c, 



\dy/x-h 



Therefore w= x(-j-} + k+hi-:-} 



1 -*"/*-h \dy/x-h 



x=h 



Example 1. To find the equations of the tangent and normal 

 to the curve y z = 4>x at the point where x = 4. 



i dy 1 

 1 hen y = 2x and -f- = 7= 



dx -yx 



when # = 4, y = 4, and -p = - 



Equation of the tangent is y = -x+ c 



2 



but 4 = - x 4 + c 



2 



and c = 2 



Then t/ - -x+ 2, or 2w = x+ 4 



2 



Equation of the normal is y = 2x+c l 



but 4= 2 x 4 + Cj 



and Cj = 12 



Then y = - 2a; + 12, or 2r + y = 12. 



Example 2. To find the equations of the tangent and normal 

 to the curve y = e* sin x. 



When # = 1, y = e sin 1 = 0-8415e 



-r- = e* sin x + e* cos x 

 = e* (sin <r + cos x) 

 When a? = 1, -1 = e(0-8415 + 0-5408) = 1-38180 





