We want to determine the graphical representant of the general bivariate quadratic equation
where are known real numbers and .
If , we will rotate the coordinate system, getting new coordinate axes and , such that the equation (1) transforms into a new one having no more the mixed term . Let the rotation angle be to the anticlockwise (positive) direction so that the - and -axes form the angles and with the original -axis, respectively. Then there is the
between the new and old coordinates (see rotation matrix). Substituting these expressions into (1) it becomes
It’s always possible to determine such that , i.e. that
for and for the case . Then the term vanishes in (2), which becomes, dropping out the apostrophes,
If none of the coefficients and equal zero, one can remove the first degree terms of (4) by first writing it as
and then translating the origin to the point , when we obtain the equation of the form
If and have the same sign (http://planetmath.org/SignumFunction), then in that (5) could have a counterpart in the plane, the sign must be the same as the sign of ; then the counterpart is the ellipse (http://planetmath.org/Ellipse2)
If and have opposite signs and , then the curve (5) correspondingly is one of the hyperbolas (http://planetmath.org/Hyperbola2)
which for is reduced to a pair of intersecting lines.
The kind of the quadratic curve (1) can also be found out directly from this original form of the equation. Namely, from the formulae (3) between the old and the new coefficients one may derive the connection
when one first adds and subtracts them obtaining
Two latter of these give
and when one subtracts this from the equation , the result is (7), which due to the choice of is simply
Thus the curve is, when it is real,
for an ellipse (http://planetmath.org/Ellipse2),
for a hyperbola (http://planetmath.org/Hyperbola2) or two intersecting lines,
for a parabola (http://planetmath.org/Parabola2) or a double line.
- 1 L. Lindelöf: Analyyttisen geometrian oppikirja. Kolmas painos. Suomalaisen Kirjallisuuden Seura, Helsinki (1924).
|Date of creation||2013-03-22 17:56:26|
|Last modified on||2013-03-22 17:56:26|
|Last modified by||pahio (2872)|
|Synonym||graph of quadratic equation|