show that the curves x3 - 3xy2 +2= 0 and 3x2y - y3 -2 =0 cut orthogonally - Maths - Application of Derivatives

Question

show that the curves x3 - 3xy2 +2= 0 and
3x2y - y3 -2 =0 cut orthogonally

Ishwarmani · Accepted Answer

Dear Student, Please find below the solution to the asked query:x3-3xy2 = -2                
1and 3x2y - y3 = 2         
2differentiating 1 both side w r
t  x,3x2-31·y2 + x·2ydydx = 0⇒x2-y2 - 2xydydx = 0∴dydx =x2-y2 2xy      
3similarly ,differentiating 2 both side w
r
t  x, we getdydx =-2xy x2-y2          
4let the curves intersect at a,bfrom3,slope of the tangent to curve 1 at  a,b i
e
 m1 =a2-b2 2abfrom4,slope of the tangent to curve2 at  a,b i
e
 m2 =-2ab a2-b2 ∴ m1m2 = a2-b2 2ab-2ab a2-b2   = -1∴two curves cut each other orthogonally
Hope this information will clear your doubts about the topic
  If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible

show that the curves x3 - 3xy2 +2= 0 and 3x2y - y3 -2 =0 cut orthogonally.

show that the curves x³- 3xy²+2= 0 and 3x²y - y³ -2 =0 cut orthogonally.