Question

For each real value of k, the equation y=kx^-1/3 defines a family of curves in the x-y plane. Find an algebraic equation that defines the family of curves that are orthogonal to the given family

Answer 1

i have to wonder if the family would have to have a negative reciprocal of the tangents of this one

Answer 2

ill assume the ^-1/3 is all an exponent

y = kx^-1/3

y' = -k/3x^4/3

flip and negate it

Y' = (3x^4/3)/k

int Y' = 3(x^7/3)/k + C

Answer 3

my idea is to define the tangents at all points for the given; and since a negative flip gives us perp slopes; i negate and flip the y' function and the integrate it back up for position