Hello, chaosier!

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 [3] - [4]: .

. . Hence: .

Substitute into [3]: .

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 [6]: .

. . Substitute into [5]: .