Question

Find the differential equation of the space curve in which the two families of
surfaces
u = x^2 − y^2 = c1 and
v = y^2 − z^2 = c2 intersect.

Answer 1

$u = x^2 − y^2 = c_1...(1) \newline v = y^2 − z^2 = c2 ...(2)$

differentiating both (1) and (2)

$du=2xdx-2ydy=0\newline \implies xdx=ydy ...(3)\newline dv=2ydy-2zdz=0\newline \implies ydy=zdz...(4) \newline by\hspace{0.1cm} (3) \hspace{0.1cm}and \hspace{0.1cm} (4)\newline\implies xdx=ydy=zdz \newline \implies \frac{xdx}{xyz}=\frac{ydy}{xyz}=\frac{zdz}{xyz}\newline \implies \frac{dx}{yz}= \frac{dy}{xz}= \frac{dz}{xy}$

Answer:$\frac{dx}{yz}= \frac{dy}{xz}= \frac{dz}{xy}$