Question

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?

Answer 1

• `( 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)`