Question
The following equation defines a family of plane curves. x3 + y3 - 3axy = c where c ( - infinity; + infinity). A unique family member passes through every point in the XY-plane except (0. 0) and (a, a). Let (alpha, beta) be a point on one of these plane curves. Your particular parameter and coordinates are a = -8 and (alpha, beta) = (-4,-8). For (alpha, beta) = (-4,-8) calculated your value of constant c. Using MATLAB's ezplot or similar software that does implicit plots, plot on the same plot, (i).x3 +y3 -3axy = 0 and (ii) x3 + y3 -3axy = c. Using implicit differentiation derive the derivative, dy/dx, for this family of curves.
Explanation / Answer
a)
c = x^3 + y^3 -3axy = (-4)^3 + (-8)^3 - 3*(-8)*(-4)*(-8) = 192
b)
Use the following code in matlab:
ezplot('x^3 + y^3 + 24xy')
and
ezplot('x^3+y^3+24xy -192')
c)
d/dx (x^3 + y^3 - 3axy = c)
3x^2 + 3y'y^2 - 3ay - 3axy' = 0
y'(3ax - 3y^2) = 3x^2 - 3ay
y' = (x^2 - ay)/(ax - y^2)
for a = -8 :
y' = (x^2 + 8y)/(-8x - y^2)