Solve $F(x,y,p)=(8p^3-27)x-12p^2y=0$
My attempt: $F=0, \frac{\partial F}{\partial p}=0,$
Solving them i got $p=\frac{y}{x}$
Now subsituting this in above, we get $\frac{x(8y^3-27x^3)}{x^3}=12 \frac{y^2}{x^2}y \implies 4y^3+27x^3=0 \rightarrow$ this is my p-discriminant. But p-descriminant solution is given as $x(4y^3+27x^3)=0$. Pls clarify
Best Answer
Hint: With derivative of $y=\dfrac{(8p^3-27)x}{12p^2}$ respect to $x$ and simplification we find $(2xp'+p)(4p^3+27)=0$ which gives us the solutions.