find the orthogonal trajectory of y = (k/x)
first we find the diff. equation of the given family
diff. w.r.t. x
y' = -k / (x²)---------(2)
eliminating k , divide (2) by (1)
y' / y = -1/x ---------(3)
(3) is the diff. equation of the given family.
replace y with -1/y' in (3) to get the diff. equation of the orthogonal trajectory
diff. equation of the orthogonal trajectory is
-yy' = -1/x
y(dy/dx) = 1/x
integrating and simplifying, we see that the orthogonal trajectory is
x² - y² = C
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