I am given equations with coefficients related by a common ratio,
thus a geometric sequence, such as: .
I plotted numerous lines which obey this geometric pattern
. . and discovered that there is a curve which,
. . through trial and error, I found to be: . . . . . I agree!
I also found that each line was tangential to the curve.
I proceeded to create a general proof of the solutions
. . of equations with this general form: .
I found the solution of and using simultaneous equations.
I am now stuck on proving the curve.
Can anybody tell me if my approach above is correct
. . and where to go from this point?
Subtract  - : .
. . Hence: .
Substitute into : .
We have parametric equations for the intersection
. . of two members of this family of curves.
. . . . .
To determine the "envelope" of this family of curves, let
. . Hence, we have: .
Eliminate the parameter:
. . From : .
. . Substitute into : .