The two curves x3−3xy2+2=0  and 3x2y−y3−2=0 intersect at an angle of whose degree measure is

# The two curves  and $3{x}^{2}y-{y}^{3}-2=0$ intersect at an angle of whose degree measure is

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### Solution:

For the curve ${x}^{3}-3x{y}^{2}+2=0$, the gradient is $\frac{dy}{dx}=-\frac{3{x}^{2}-3{y}^{2}}{-6xy}=\frac{{x}^{2}-{y}^{2}}{2xy}$

For the curve $3{x}^{2}y-{y}^{3}-2=0$, the gradient is $\frac{dy}{dx}=-\frac{6xy}{3{x}^{2}-3{y}^{2}}=-\frac{2xy}{{x}^{2}-{y}^{2}}$

Observing the above two gradients, the product of slopes is -1

hence, the angle between the curves is ${90}^{0}$

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