Algebra II: Coordinate Geometry

On this page we hope to clear up any problems you might have with coordinate geometry (circles, ellipses, midpoints, etc.). If you're in to graphs, this is your section!

Distance and midpoint formulas
Circles
Ellipses
Hyperbolas
Systems of equations
Quiz on coordinate geometry

Distance and Midpoint

When dealing with lines and points, it is very important to be able to find out how long a line segment is or to find a midpoint. However, since the midpoint and distance formulas are covered in most geometry courses, you can follow this link to better you understanding of the midpoint and distance formulas.

Circles

Circles, when graphed on the coordinate plane, have an equation of x^2 + y^2 = r^2 where r is the radius (standard form) when the center of the circle is the origin. When the center of the circle is (h, k) and the radius is of length r, the equation of a circle (standard form) is (x - h)^2 + (y - k)^2 = r^2.

Example:

1.  Problem: Find the center and radius of (x - 2)^2 + (y + 3)^2 = 16.
             Then, graph the circle.
   Solution: Rewrite the equation in standard form.

             (x - 2)^2 + [y - (-3)]^2 = 4^2

             The center is (2, -3) and the radius is 4.  The graph
             is easy to draw, especially if you use a compass.

             The accompanying figure shows a graph
             of the solution.

Ellipses

Ellipses, or ovals, when centered at the origin, have an equation (standard form) of ((x^2)/(a^2)) + ((y^2)/(b^2)) = 1. When the center of the ellipse is at (h, k), the equation (in standard form) is as follows:

            (x - h)^2   (y - k)^2
            --------- + --------- = 1
               a^2         b^2

Example:

1.  Problem: Graph x^2 + 16y^2 = 16.
   Solution: Multiply both sides by 1/16 to put the equation in
             standard form.

             x^2   y^2
             --- + --- = 1
              16    1 

             a = 4 and b = 1.  The vertices are at
             (±4, 0) and (0, ±1).  The points are on
             the axes because the equation tells us the center is at the origin,
             so the vertices have to be on the axes).

             Connect the vertices to form an oval, and you are done!  See the
             accompanying figure to see a graph
             of the solution.

Hyperbolas

The equation of a hyperbola (in standard form) centered at the origin is as follows:

            x^2   y^2
            --- - --- = 1
            a^2   b^2

Example:

1.  Problem: Graph 9x^2 - 16y^2 = 144.
   Solution: First, multiply each side of the equation by 1/144 to put
             it in standard form.

             x^2   y^2
             --- - --- = 1
              16    9 
   
             We know that a = 4 and b = 3.  The vertices are at
             (±4, 0).  (Since we know the center is at the origin,
             we know that the vertices are on the x-axis.)

             The easiest way to graph a hyperbola is to draw a rectangle using
             the vertices and b, which is on the y-axis.

             Draw asymptotes through opposite corners of the rectangle.

             Next, draw the hyperbola.  The accompanying figure is
             the graph of 9x^2 - 16y^2 = 144.

Systems of Equations

The easiest way to solve systems of equations that include circles, ellipses, or hyperbolas, is graphically. Because of the shapes (circles, ellipses, etc.), there can be more than one solution.

Example:

1.  Problem: Solve the following system of equations:
             x^2 + y^2 = 25
             3x - 4y = 0
   Solution: Graph both equations on the same coordinate plane.  The points
             of intersection have to satisfy both equations, so be sure to 
             check the solutions.  (Both intersections do check.)

             This example figure shows the
             graphs and intersections (solutions) of both equations.

Take the quiz on coordinate geometry. The quiz is very useful for either review or to see if you've really got the topic down.

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