**ANALYTICAL GEOMETRY**

**Areas of focus:**

- Commonly used coordinate systems
- Equation of a straight line
- Equation of a circle
- Equation of conic sections

**Commonly
used coordinate systems:**

**Rectangular coordinates in 2-D:**

The location of a point in a two-dimensional plane
can be represented by a pair of numbers representing the coordinates of the
point in a rectangular coordinate system. For example, the point *A* in
the figure has coordinates (*x, y*) in the rectangular *x-y*
coordinate system shown. To determine a coordinate one draws a perpendicular
onto the coordinate axis. In this case, the arrows on the coordinate axis
indicate that points to the right of the origin *O* on the *x*-axis
are positive and to the left are negative. In a similar manner, points above
the origin on the *y*-axis are positive and below it are negative.

**Polar coordinates:**

The location of a point in a two dimensional plane
can be represented by a pair of numbers representing the coordinates of the
point in a polar coordinate system. For example, the point *A* in the
figure has coordinates in the polar coordinate system shown. In this
coordinate system *r* represents the radial distance from the reference
point *O* to the point *A* and represents the angle the line *OA*
makes with the reference line *OB*. As indicated by the arrow, the angle
is positive if measured counter clockwise from *OB* and negative if
measured clockwise.

**Rectangular coordinates in 3-D:**

The location of a point in three-dimensional space
can be represented by a triplet of numbers representing the coordinates of the
point in a rectangular coordinate system. For example, point *A* in the
figure has coordinates (*x, y, z*) in the rectangular coordinate system
shown. To get these coordinates one can drop a perpendicular line from *A*
onto the *x-y* plane to get point *B* and then draw perpendiculars
onto the *x-* and *y*-axes.

**Cylindrical Coordinates:**

** **

The coordinates of a point in three-dimensional space can be represented in cylindrical coordinates by the triplet () as shown in the figure.

**Spherical Coordinates:**

The coordinates of a point in three-dimensional space can be represented in spherical coordinates by the triplet as shown in the figure.

The equation of a straight line in a plane is given in the *x-y* coordinate
system by the set of points (*x, y*) that satisfy the equation

where, as shown in the figure, *a* represents the slope of the line in
terms of its rise divided by its run, and *b* is the *y*-coordinate
of the point of intercept of the line and the *y*-axis.

One can evaluate the equation of a line from any two points on it. For
example, consider points *A* and *B* shown in the figure with
coordinates (x_{1},y_{1}) and (x_{2},y_{2}),
respectively. Since ACD and ABE are similar triangles, we have

This can be put in the above format by selecting

If x_{1} = 0, then, as can be seen from the figure,

** **

The circle centered at the origin of a rectangular coordinate system is
given by the set of all points (*x,y*) that satisfy the equation

where, as can be seen in the figure, *r* is the radius of the circle.

In polar coordinates the equation of a circle is given by specifying the
radial coordinate *r* to be constant.

For a circle centered at point (*x*_{1},*y*_{1})
and of radius *r*, the equation of the points on the circle is given by

The ellipse, parabola, and hyperbola are conic sections. Their curves have
the distinct characteristic that each point on the curve is such that the ratio
of its distance from a line known as the directrix and a point known as the
focus is a given constant. In the figure, *F* is the focus, *AB* is
the directrix and, the constant *e*, known as the eccentricity, defines
the conic section and is given by

** **

The value of the eccentricity defines the conic shape.

*e *< 1 gives an ellipse

e = 1 gives a parabola

e > 1 gives a hyperbola

** **

**Equation of an ellipse:**

The equation of an ellipse in the rectangular *x-y*
coordinate system is given by

where *a* and *b* are half the lengths,
respectively, of the major and minor axis.

**Equations of parabolas:**

The basic equation of a parabola in a rectangular *x-y*
coordinate system is given by

Depending on the sign of *a*, this equation
will result in one of the two following graphs.

A parabola that is rotate 90^{o} can be
represented by the equation

The graph of this is shown in the following figure.

The equation for parabolas moved from the origin to another point is given in the following figures.

**Equations of hyperbolas:**

** **

The equation of a hyperbola centered at the origin
with its major axis on the *x*-axis is given by

The figure shows the graph of this equation, where
F_{1} and F_{2} are the focal points and the major axis is the
line that passes through these points.

The lengths of the major and minor axis of the
hyperbola are given by *2a* and *2b*, respectively, as shown in the
following figure.

If the center of the hyperbola is moved from the
origin to a point with coordinates (*x*_{1},*y*_{1}),
then the equation of the hyperbola becomes

To get a hyperbola with major axis parallel to the
vertical axis, one can change *x* and *y* in the equations.

A hyperbola with equal major and minor axis and
with axis rotated 45^{o} from the *x-y* axes has an equation

where the major and minor axes each have a length of . The figure shows this hyperbola.