Show that $P(x,y)=0$ is a hyperbola if $b^2−4ac>0$ .
The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach.
I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. And I stuck!
Any hint to start solving this?
proof-verification algebraic-geometry conic-sections plane-curves
asked Nov 24 at 11:28
72D
564116
add a comment |
The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach.
I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. And I stuck!
Any hint to start solving this?
proof-verification algebraic-geometry conic-sections plane-curves
asked Nov 24 at 11:28
72D
564116
Isn’t this more or less the same problem that you’ve already attacked in previous questions with different values of the discriminant?
– amd
Nov 24 at 22:51
1
@amd, Some remarks: 1- I am a member of MSE for less than a month, and my experience to this day is that anytime I've tried to ask only a little more/different question I've been replied like why don't you ask it as a new post? And this question would've been much longer if I had posted all three discriminant cases in one single question so most probably even worse replies.. Also, answers (= resulting sets to be explained) are different. 2- I introduced $Delta_x(y)$, etc for better clarifications because may not anyone who reads this question reads also any other previous questions.
– 72D
Nov 24 at 23:06
add a comment |
The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach.
I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. And I stuck!
Any hint to start solving this?
proof-verification algebraic-geometry conic-sections plane-curves
asked Nov 24 at 11:28
72D
564116
The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach.
I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. And I stuck!
Any hint to start solving this?
proof-verification algebraic-geometry conic-sections plane-curves
proof-verification algebraic-geometry conic-sections plane-curves
asked Nov 24 at 11:28
72D
564116
asked Nov 24 at 11:28
72D
564116
asked Nov 24 at 11:28
72D
564116
asked Nov 24 at 11:28
72D
564116
asked Nov 24 at 11:28
72D
564116
564116
Isn’t this more or less the same problem that you’ve already attacked in previous questions with different values of the discriminant?
– amd
Nov 24 at 22:51
1
@amd, Some remarks: 1- I am a member of MSE for less than a month, and my experience to this day is that anytime I've tried to ask only a little more/different question I've been replied like why don't you ask it as a new post? And this question would've been much longer if I had posted all three discriminant cases in one single question so most probably even worse replies.. Also, answers (= resulting sets to be explained) are different. 2- I introduced $Delta_x(y)$, etc for better clarifications because may not anyone who reads this question reads also any other previous questions.
– 72D
Nov 24 at 23:06
add a comment |
Isn’t this more or less the same problem that you’ve already attacked in previous questions with different values of the discriminant?
– amd
Nov 24 at 22:51
1
@amd, Some remarks: 1- I am a member of MSE for less than a month, and my experience to this day is that anytime I've tried to ask only a little more/different question I've been replied like why don't you ask it as a new post? And this question would've been much longer if I had posted all three discriminant cases in one single question so most probably even worse replies.. Also, answers (= resulting sets to be explained) are different. 2- I introduced $Delta_x(y)$, etc for better clarifications because may not anyone who reads this question reads also any other previous questions.
– 72D
Nov 24 at 23:06
Isn’t this more or less the same problem that you’ve already attacked in previous questions with different values of the discriminant?
– amd
Nov 24 at 22:51
Isn’t this more or less the same problem that you’ve already attacked in previous questions with different values of the discriminant?
– amd
Nov 24 at 22:51
1
1
@amd, Some remarks: 1- I am a member of MSE for less than a month, and my experience to this day is that anytime I've tried to ask only a little more/different question I've been replied like why don't you ask it as a new post? And this question would've been much longer if I had posted all three discriminant cases in one single question so most probably even worse replies.. Also, answers (= resulting sets to be explained) are different. 2- I introduced $Delta_x(y)$, etc for better clarifications because may not anyone who reads this question reads also any other previous questions.
– 72D
Nov 24 at 23:06
@amd, Some remarks: 1- I am a member of MSE for less than a month, and my experience to this day is that anytime I've tried to ask only a little more/different question I've been replied like why don't you ask it as a new post? And this question would've been much longer if I had posted all three discriminant cases in one single question so most probably even worse replies.. Also, answers (= resulting sets to be explained) are different. 2- I introduced $Delta_x(y)$, etc for better clarifications because may not anyone who reads this question reads also any other previous questions.
– 72D
Nov 24 at 23:06
add a comment |
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Isn’t this more or less the same problem that you’ve already attacked in previous questions with different values of the discriminant?
– amd
Nov 24 at 22:51
1
@amd, Some remarks: 1- I am a member of MSE for less than a month, and my experience to this day is that anytime I've tried to ask only a little more/different question I've been replied like why don't you ask it as a new post? And this question would've been much longer if I had posted all three discriminant cases in one single question so most probably even worse replies.. Also, answers (= resulting sets to be explained) are different. 2- I introduced $Delta_x(y)$, etc for better clarifications because may not anyone who reads this question reads also any other previous questions.
– 72D
Nov 24 at 23:06