Originally Posted by
ticbol I don't know if you have studied yet all those: line, parabola, ellipse, hyperbola, circle. If yes, then you may understand my answer. If not all of them yet, then you will not fully understand my answer.
Line.
It is linear or of degree 1. Meaning, the x or the y have exponents of 1 only.
It is in the form, y = Ax +B. Or, Ax +By +C = 0.
Parabola.
It is quadratic, or of degree 2. One of the variables is of degree 1 only, though.
It is in the expanded form, y = Ax^2 +Bx +C. Or x = Ay^2 +By +C.
Circle.
It is quadratic, or of degree 2. Both variables are in degree 2. The coefficients of the two squared variables are equal or the same. It is the sum of the two squared variables, etc.
In its expanded form, Ax^2 +Ay^2 +Bx +Cy +D = 0
See that Ax^2 + Ay^2 ?
Ellipse.
It is quadratic, or of degree 2. Both variables are in degree 2. The coefficients of the two squared variables are unequal or not the same. It is the sum of the two squared variables, etc.
In its expanded form, Ax^2 +By^2 +Cx +Dy +E = 0
See that Ax^2 + By^2 ?
Hyperbola.
It is quadratic, or of degree 2. Both variables are in degree 2. The coefficients of the two squared variables may be equal or not equal. It is the subtraction of the two squared variables, etc.
In its expanded form, Ax^2 -Ay^2 +Bx +Cy +D = 0
Or, Ax^2 -By^2 +Cx +Dy +E = 0
See those Ax^2 -Ay^2, or Ax^2 -By^2 ?
----Sometimes it has the "xy" term in the expanded form, even if the squared variables as not in there. Meaning, if you see a quadratic equation with an "xy" term, sum or subtraction, the equation is hyperbola.
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So, the two equations in your question here are both circles, although the last one is imaginary because the sum of two squared quantities (after completing the squares) cannot be negative. --------answer.