Homework SourseFor each real value of k the equation y kx13defines a famil
For each real value of k, the equation y = kx-1/3defines a family of curves
in the xy-plane. Find an algebraic equation that defines the family of curves
which are orthogonal to the given family.
in the xy-plane. Find an algebraic equation that defines the family of curves
which are orthogonal to the given family.
Solution
k is a constant so we have:
y = kx - 1/3
(y + 1/3) = kx
(y+1/3)/x = k
Now differentiate both sides:
[(y\'+0)*x - 1*(y + 1/3)]/x2 = 0
y\'x - (y+1/3) = 0
So the differentil equation is:
y\' = (y+1/3)/x
So the equation of the orthogonal curve is:
y\' = -1/((y+1/3)/x) = -x/(y + 1/3)
