The following equation defines a family of plane curves. x3 + y3 - 3axy = c wher

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)

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