Find the differential equations of the space curve in which the two families of surfaces #u=x^2-y^2=c_1# and #v=y^2-z^2=c_2# intersect?

1 Answer
Jul 4, 2018

Answer:

  • # ( dx)/(yz) = ( dy)/(xz)= ( dz)/(xy)#

Explanation:

Because #u(bbx)# and #v(bbx)# are level surfaces: #du = dv = 0#

  • #du = underbrace(2x)_(u_x)dx - underbrace(2y)_(u_y)dy = 0 qquad implies x dx = y dy qquad bbbA#

Likewise;

  • #dv = 2ydy - 2zdz = 0 qquad implies y dy = z dz qquad bbbB#

#bbbA" & "bbbB implies x dx = y dy= z dz#

#:. (x dx)/(xyz) = (y dy)/(xyz)= (z dz)/(xyz)#

  • # ( dx)/(yz) = ( dy)/(xz)= ( dz)/(xy)#