The curves ` x^(3) -3xy^(2) = a and 3x^(2)y -y^(3)=b,` where a and b are constants, cut each other at an angle of

Explanation

Differentiation `x^(3)-3xy^(2)=a` w.r.t. x we obtain
`3x^(2)-3(x.2y""(dy)/(dx)+1.y^(2))=0`
`implies x^(2)-y^(2)-2xy""(dy)/(dx)=0`
`:. (dy)/(dx)=(x^(2)-y^(2))/(2xy)` ..... (1)
Again differentiating ` 3x^(2)y-y(3)=b` w.r.t. , we get
`3(2xy+x^(2)""(dy)/(dx))-3y^(2)""(dy)/(dx)=0 `
`implies (x^(2)-y^(2))""(dy)/(dx)+2xy=0`
`:. (dy)/(dx)=-(2xy)/(x^(2)-y^(2))`........(2)
Let the two curves intersect at `P(x_(0),y_(0))` . From (1) at P
`((dx)/(dy))_(1)=(x_(0)^(2)-y_(0)^(2))/(2x_(0)y_(0))` for curve (1)
`((dy)/(dx))_(2)=-(2x_(0)y_(0))/(x_(0)^(2)-y_(0)^(2)) ` for curve (2)
Since `((dy)/(dx))_(1).((dy)/(dx))_(2)=-1`, the curves ` x^(3)-3xy^(2)=a and 3x^(2)y-y^(3)=b` cut each other at right angle , i.e., they are orthogonal.