Prove that the curves `y=x^(3) and xy=k` cut each other orthogonally, if 3k= 1.

Explanation

Equation of curves
`y=x^(3)….(1) " "xy = k ….(2)`
`rArr (dy)/(dx) = 3 x^(2) " "rArr x (dy)/(dx) + y = 0`
`rArr (dy)/(dx) = 3x^(2) " "rArr (dy)/(dx) =-y/x`
Let the point of intersection be `(x_(1), y_(1))`.
`:." Slope of tangents of the curve at "(x_(1), y_(1))" are "m_(1) = 3x_(1)^(2) and m_(2) =-(y_(1))/(x_(1))*`
`:'` The curves orthogonally.
`:. m_(1) m_(2) =-1`
` rArr 3x_(1)^(2) (-(y_(1))/(x_(1)))=-1" "rArr 3x_(1)y_(1) = 1 ....(3)`
From eq.(2)
`x_(1)y_(1) = k`
Put this value in eq.(3)
3k = 1.