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One dimension: Points
A point is specified by a single real number called its
coordinate. Let the coordinate of P1 be
x1, and that of P2 be x2.
The distance from P1 to P2 is
d = sqrt[(x1-x2)2] = |x1-x2|.
The coordinate of the point dividing the line segment
P1P2 in the ratio r/s is
[rx2+sx1]/[r+s]. As a special case, when r = s,
the midpoint of the line segment has coordinate
The set of all points with coordinate x satisfying a linear equation in
x is a single point. Its equation has the general form
Ax + B = 0,
where A is nonzero. The coordinate of the point is -B/A.
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One dimension: General Quadratic Equations
A general quadratic equation can be put into the following form:
ax2 + 2bx + c = 0,
where a is nonzero. If we have an equation of this kind, it can
represent one of three different kinds of curve. Which kind depends on
the value of the following quantity:
The cases are as follows:
The equation can be put into standard form by completing the square,
and then making a translation to move the center to the origin:
Case sign(Delta) Name Standard form
1 - Real circle x2/r2 = 1
2 + Imaginary circle x2/r2 = -1
3 0 Coincident lines x2 = 0
0 = ax2 + 2bx + c,
0 = a2x2 + 2abx + ac,
0 = (ax + b)2 - b2 + ac,
-Delta = (ax + b)2,
-Delta/a2 = y2,
where y = x + b/a,
-sign(Delta) = y2/r2,
where r = sqrt(|Delta|)/a, provided Delta is nonzero. The center of
the circle has coordinate -b/a, and the radius is r.
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One dimension: Circles
A circle is the set of all points at a distance (or radius) r > 0 from
the center P1. Its equation has the form
(x-x1)2 = r2,
|x-x1| = r.
It consists of just two points, whose coordinates are x1 + r
and x1 - r.