Download Analytical Geometry-Mathmatics-Lecture Handoit and more Lecture notes Mathematics in PDF only on Docsity!
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.
The coordinates of a point in three-dimensional space can be represented in spherical coordinates by the triplet as shown in the figure.
Equation of a straight line:
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 ,y1) and (x 2 ,y2 ), 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,
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
Equations of conic sections:
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
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
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 45o from the x-y axes has an equation
where the major and minor axes each have a length of. The figure shows this hyperbola.
