Show that $P(x,y)=0$ is a parabola if $b^2-4ac=0$.











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I tried the following:



I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. Letting $b^2-4ac=0$ makes $Delta_x(y)$ a linear function of $y$ and if $y ge frac{4ah -d^2}{2bd − 4ae}$ then for any of those $y$'s there is only one corresponding $x$. So IF for the $P(x, y)$ there are only four possible shapes: circle, ellipse, parabola and hyperbola then the only choice is parabola.



But the proof is not only complete but also doesn't look so rigorous. Are there any better proofs?










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  • Strictly speaking, $P(x,y)=0$ is not necessarily a parabola when $b^2-4ac=0$. Take $a=b=c=d=0,e=h=1$. But as I've written in the answer to your another question, we can say that by a suitable rotation, $P(x,y)=0$ becomes an equation of the from $AX^2+CY^2+DX+EY+F=0$ with $AC=0$.
    – mathlove
    Nov 27 at 5:26












  • @mathlove, evaluation of P(x,y)=0 by rotation method is much easier, but the book didn't introduced it (yet?). But after learning about it in your answer I could evaluated different cases too.
    – 72D
    Nov 27 at 5:50















up vote
0
down vote

favorite












I tried the following:



I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. Letting $b^2-4ac=0$ makes $Delta_x(y)$ a linear function of $y$ and if $y ge frac{4ah -d^2}{2bd − 4ae}$ then for any of those $y$'s there is only one corresponding $x$. So IF for the $P(x, y)$ there are only four possible shapes: circle, ellipse, parabola and hyperbola then the only choice is parabola.



But the proof is not only complete but also doesn't look so rigorous. Are there any better proofs?










share|cite|improve this question
























  • Strictly speaking, $P(x,y)=0$ is not necessarily a parabola when $b^2-4ac=0$. Take $a=b=c=d=0,e=h=1$. But as I've written in the answer to your another question, we can say that by a suitable rotation, $P(x,y)=0$ becomes an equation of the from $AX^2+CY^2+DX+EY+F=0$ with $AC=0$.
    – mathlove
    Nov 27 at 5:26












  • @mathlove, evaluation of P(x,y)=0 by rotation method is much easier, but the book didn't introduced it (yet?). But after learning about it in your answer I could evaluated different cases too.
    – 72D
    Nov 27 at 5:50













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I tried the following:



I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. Letting $b^2-4ac=0$ makes $Delta_x(y)$ a linear function of $y$ and if $y ge frac{4ah -d^2}{2bd − 4ae}$ then for any of those $y$'s there is only one corresponding $x$. So IF for the $P(x, y)$ there are only four possible shapes: circle, ellipse, parabola and hyperbola then the only choice is parabola.



But the proof is not only complete but also doesn't look so rigorous. Are there any better proofs?










share|cite|improve this question















I tried the following:



I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. Letting $b^2-4ac=0$ makes $Delta_x(y)$ a linear function of $y$ and if $y ge frac{4ah -d^2}{2bd − 4ae}$ then for any of those $y$'s there is only one corresponding $x$. So IF for the $P(x, y)$ there are only four possible shapes: circle, ellipse, parabola and hyperbola then the only choice is parabola.



But the proof is not only complete but also doesn't look so rigorous. Are there any better proofs?







proof-verification algebraic-geometry conic-sections plane-curves






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 22 at 13:38

























asked Nov 22 at 12:26









72D

536116




536116












  • Strictly speaking, $P(x,y)=0$ is not necessarily a parabola when $b^2-4ac=0$. Take $a=b=c=d=0,e=h=1$. But as I've written in the answer to your another question, we can say that by a suitable rotation, $P(x,y)=0$ becomes an equation of the from $AX^2+CY^2+DX+EY+F=0$ with $AC=0$.
    – mathlove
    Nov 27 at 5:26












  • @mathlove, evaluation of P(x,y)=0 by rotation method is much easier, but the book didn't introduced it (yet?). But after learning about it in your answer I could evaluated different cases too.
    – 72D
    Nov 27 at 5:50


















  • Strictly speaking, $P(x,y)=0$ is not necessarily a parabola when $b^2-4ac=0$. Take $a=b=c=d=0,e=h=1$. But as I've written in the answer to your another question, we can say that by a suitable rotation, $P(x,y)=0$ becomes an equation of the from $AX^2+CY^2+DX+EY+F=0$ with $AC=0$.
    – mathlove
    Nov 27 at 5:26












  • @mathlove, evaluation of P(x,y)=0 by rotation method is much easier, but the book didn't introduced it (yet?). But after learning about it in your answer I could evaluated different cases too.
    – 72D
    Nov 27 at 5:50
















Strictly speaking, $P(x,y)=0$ is not necessarily a parabola when $b^2-4ac=0$. Take $a=b=c=d=0,e=h=1$. But as I've written in the answer to your another question, we can say that by a suitable rotation, $P(x,y)=0$ becomes an equation of the from $AX^2+CY^2+DX+EY+F=0$ with $AC=0$.
– mathlove
Nov 27 at 5:26






Strictly speaking, $P(x,y)=0$ is not necessarily a parabola when $b^2-4ac=0$. Take $a=b=c=d=0,e=h=1$. But as I've written in the answer to your another question, we can say that by a suitable rotation, $P(x,y)=0$ becomes an equation of the from $AX^2+CY^2+DX+EY+F=0$ with $AC=0$.
– mathlove
Nov 27 at 5:26














@mathlove, evaluation of P(x,y)=0 by rotation method is much easier, but the book didn't introduced it (yet?). But after learning about it in your answer I could evaluated different cases too.
– 72D
Nov 27 at 5:50




@mathlove, evaluation of P(x,y)=0 by rotation method is much easier, but the book didn't introduced it (yet?). But after learning about it in your answer I could evaluated different cases too.
– 72D
Nov 27 at 5:50















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