Theory of General Conics

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Unit 13

• Theory of General Conics

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13.1 Reduction of Equations of Quadratic Curves
to Standard Forms
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13.1 Reduction of Equations of Quadratic Curves
to Standard Forms
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13.1 Reduction of Equations of Quadratic Curves
to Standard Forms
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13.1 Reduction of Equations of Quadratic Curves
to Standard Forms
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13.1 Reduction of Equations of Quadratic Curves
to Standard Forms
Any equation of a quadratic curve can be reduced
to a standard equation of a conic by the
following steps of transformation of coordinate
axes
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13.1 Reduction of Equations of Quadratic Curves
to Standard Forms
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P.472 Ex. 13A
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13.2 Invariants
The coefficients of the quadratic terms (i.e. x2,
y2 and xy terms) of the equation of a quadratic
curve Ax2 Bxy Cy2 Dx Ey F0 are
invariants under translation of coordinate axes.
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13.2 Invariants
A C and B2 4AC of the equation of the
quadratic curve Ax2 Bxy Cy2 Dx Ey
F0 are invariants under rotation of coordinate
axes about the origin.
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13.2 Invariants
A C and B2 4AC of the equation of the
quadratic curve Ax2 Bxy Cy2 Dx Ey
F0 are invariants under the transformation of
coordinate axes.
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13.2 Invariants
The graph of the quadratic equation Ax2 Bxy
Cy2 Dx Ey F0 is a parabola, an ellipse, a
hyperbola, a circle, a point, or a pair of
straight lines.
Since ellipse and hyperbola both have a centre,
they are called central conics. On the contrary,
parabola, which has no centre, is known as
non-central conic.
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13.2 Invariants
A circle, a single point and a pair of straight
lines are particular cases of conics. A single
point is known as degenerated ellipse. A pair of
intersecting straight lines is known as
degenerated hyperbola while a pair of parallel
straight lines or coincident straight lines is a
degenerated parabola.
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13.2 Invariants
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13.2 Invariants

• Remarks
• The sign of B2 - 4AC determines the type of curve
  that an equation of second degree. This
  expression is quite easy to be memorized since it
  is analogous to the discriminant, b2 - 4ac, of a
  quadratic equation ax2 bx c 0.
• To determine the type of curve that an equation
  of second degree represents, first we have to
  check whether it represents a pair of straight
  lines or not and then check the sign of B2 4AC
  for conics.

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P.478 Ex. 13B