Coordinate Geometry

Excerpts

• Coordinate Geometry

  This conception, simple as it looks, is the main idea of the great subject of coordinate geometry.

• Coordinate Geometry

  A locus is the curve (or surface, if we do not confine ourselves to a plane) formed by points, all of which possess some given property.

• Coordinate Geometry

  Consider $y - x = 1$: the corresponding locus does not pass through the origin. We therefore seek where it cuts the axes.

• Coordinate Geometry

  We each of us refer our sensible perceptions of things to an origin which we call “here”: our location in a particular part of space round which we group the whole Universe is the essential fact of our bodily existence.

• Coordinate Geometry

  Its discovery marks a momentous epoch in the history of mathematical thought.

• Coordinate Geometry

  Euclid always contemplates a straight line as drawn between two definite points, and is very careful to mention when it is to be produced beyond this segment.

• Coordinate Geometry

  Variables, like $a$, $b$, and $c$ above, which are used to determine the correlation are called “constants,” or parameters.

Equations

• This equation is in Variables (Variables)

• Coordinate Geometry

  x - y = 1

  A second example of a correlation between the variable numbers x and y.

• Coordinate Geometry

  ax + by = c

  The general linear correlation between x and y, in which a, b and c are constants (parameters) determining the correlation; it stands for the general algebraic form of a linear relation.

• Coordinate Geometry

  x^{2} + y^{2} = 1

  A correlation between x and y of the form of a circle centred at the origin with unit radius, the starting point for generalizing to quadratic algebraic forms.

• Coordinate Geometry

  ax^{2} + by^{2} = c

  The generalization of x^2 + y^2 = 1 with constant coefficients a, b and right-hand side c, a further algebraic form of correlation between x and y.

• Coordinate Geometry

  ax^{2} + 2hxy + by^{2} = c

  A further generalization of the quadratic correlation, adding a cross term in xy with constant coefficient 2h.

• Coordinate Geometry

  ax^{2} + 2hxy + by^{2} + 2gx + 2fy = c

  The most general of the quadratic algebraic forms listed, with linear terms added to the quadratic terms, a correlation between x and y indicating a variable correlation of a given algebraic form.

• Coordinate Geometry

  y - x = 0

  The equation of a straight line through the origin O that bisects the angle XOY; obtained from ax + by = c with a = -1, b = 1, c = 0.

• Coordinate Geometry

  y + x = 0

  The equation of a straight line through the origin that bisects the angle X'OY, the line L1OL1' of the diagram.

• Coordinate Geometry

  ax + by = 0

  The general form of the equation of any straight line through the origin.

• Coordinate Geometry

  y - x = 1

  The equation of a straight line not passing through the origin; it meets the axis OX at (1, 0) and the axis OY at (0, 1), and is parallel to LOL'.

• This equation is in Variables (Variables)

Problems

No exercises in this chapter.