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So, I was doing this exercise and it says that to find the axis of symmetry of a general parabola $$ax^2+bxy+cy^2+dx+ey+f=0$$ I can just take the partial derivative of this conic.

As I didn't understand why this works, I tried doing the derivative on a normal parabola (the one that has the axis parallel to the x or y axis $$ ax^2+bx+c=0 $$) and it works as well. Can someone explain why this works?

randomdude
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  • A partial derivative can only be performed for a function of two or more variables. I'm assuming the function you are referring to when you say "general parabola" is $y=x^2$. How can you take the partial derivative of this? – Zo-Bro-23 Jan 06 '23 at 13:03
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    If you mean the derivative of the function, it is $2x$. How can you use this to find the axis of symmetry of $x=0$? Please elaborate on how this method uses the derivative of a quadratic function to find its axis of symmetry. – Zo-Bro-23 Jan 06 '23 at 13:04
  • Its conic that has the axis of symmetry rotated to an angle, so the equation isn't just the normal equation of a parabola – randomdude Jan 06 '23 at 13:06
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    Partial derivatives yield tangent vectors. By setting the tangent vector to zero, you're looking for stationary points. When the stationary points form a line, you have an axis of symmetry around that line. – Blabbo the Verbose Jan 06 '23 at 15:07
  • @BlabbotheVerbose But if I have only one point why does it have to be axis of symmetry? – randomdude Jan 06 '23 at 15:18
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    The question unclear is written, but is a likely duplicate of this one. – Moishe Kohan Jan 06 '23 at 15:36
  • I edited once again, it would help if someone would explain why equaling the derivative of $$ ax^2+bx+c=0 $$ to 0 gives me the axis of symmetry – randomdude Jan 06 '23 at 15:51
  • @NeaguCristian: I'm pretty sure you want $y = ax^2 + bx + c$ as your vertical parabola. Please think it through. The equation $ax^2 + bx + c = 0$ gives a quadratic in $x$ alone, not the equation of a curve. Implicit differentiation will give you an axis of symmetry of the conic. (Some conics have more than one, e.g. a circle.) – hardmath Jan 06 '23 at 16:04

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Note that taking the derivative of the parabola at $y=0$ will result in a tangent line at the vertex of the curve. Since the axis of symmetry intersects this line, all you need to do is set $x$ equal to that to get its equation.

user1134699
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