Find the angle of intersection of the curves ` x^(3)-3xy^(2)=a and 3x^(2)y-y^(3)=b`

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(pi)/(2)`
D. none of these

Answer

C

Explanation

The equation of the two curves are
`C_(1) : x^(3) -3xy^(2) =a " " ` …(i)
`C_(2): 3x^(2)y -y^(3)=b " " ` …(ii)
Differentiating (i) and (ii) w.r.t. x, we get
`((dy)/(dx))_(C_(1))=(x^(2)-y^(2))/(2xy) and ((dy)/(dx))_(C_(2)) = -(2xy)/(x^(2)-y^(2))`
Clearly, `((dy)/(dx))_(C_(1)) xx((dy)/(dx))_(C_(2))= -1`
So, the two curves intersect at right angle.