The graph of an equation in two variables (x and y), is the set of all
points (x,y) that satisfy the equation.
To begin our investigation, we will consider when a = 0, 1, 2, and 3.
From this graph, four shapes are recognized.
1) When a = 0 (red), the distance formula tells us that the graph of x^2 + y^2 = 9 is a circle centered at the origin with a radius of 3. More generally, the graph of the equation is (x-h)^2 + (y-k)^2 = r^2 , with radius r and center (h,k). When h and k are not both 0 (at the origin), the circle is translated h units right or left and k units up or down.
2) When a = 1 (green) , the graph of x^2 + xy + y^2 = 9 is an ellipse. The conventional form for an ellipse would be:
Since the ellipse is rotated 45 degrees from the axis, there is an xy term. Using the formulas for the rotation of axes:
3) When a = 2 (blue) , the graph of x^2 + 2xy + y^2 = 9
is two parallel lines. This becomes obvious when you factor.
These are equations of 2 lines with y intercept at 3 or -3 and slope
(m) at -1.
4) When a = 3 (tan), the graph of x^2 + 2xy + y^2 = 9 is a hyperbola. The standard form for a hyperbola is:
This introduces the same situation as with the ellipse, in that the rotation
causes an xy term.
make you wonder what happens inbetween the values a = 0, 1, 2
This indicates that 0 and 2 are boundaries.
This also indicates that 2 is a boundary.
What happens when a is negative?
Using the coefficents a = 0, -1, -2 and -3 , we see that the original graphs are reflected across the x or y axis ( or rotated 90 degrees) but hold the same characteristics.
we find that the boundaries are still 0, 2, and -2. The radius only changes
the x and y intercept. All values of c for: