Find the equation of family of curves which intersect the family of curves xy=c at an angle `45^(@)`.

Explanation

We have family of curves of xy=c
`therefore y+x(dy)/(dx)=0` or `(dy)/(dx) =-y/x=m_(1)` (say)
Let slope of the required family of curves be `(dy)/(dx)=m_(2)`.
According to the question,
`tanpi/4= +-(m_(1)-m_(2))/(1+m_(1)m_(2))`
`rArr 1-y/x(dy)/(dx)=-y/x-(dy)/(dx)` (considering + sign)
`rArr 1+y/x=(dy)/(dx)(y/x-1)`
`rArr (x+y)dx=(y-x)dy`
`rArr xdy+ydx=ydy-xdx`
`rArr d(xy)=ydy-xdx`
Integrating both sides, we get
`xy=y^(2)/2-x^(2)/2+c`, which is required family of curves.