I am given equations with coefficients related by a common ratio, thus a geometric sequence, such as

2x+4y=8

3x+12y=48

I plotted numerous lines which obey this geometric pattern and discovered that there is a curve which through trial and error I discovered to be y=(-4x)^(1/2). I also discovered that each line was tangential to the curve.

I proceeded to create a general proof of the solutions of equations obeying this general form:

ax+ary=ar^2

bx+bdy=bd^2

I discovered the solution of x and y using simultaneous equations (I am not posting the answer here because this is a portfolio task and direct answers are not allowed).

I am now stuck on proving the curve; can anybody tell me if my approach above is correct in working the relations and to go where from this point.