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I am given equations with coefficients related by a common ratio,

thus a geometric sequence, such as: .$\displaystyle \begin{array}{ccc}2x+4y&=&8 \\ 3x+12y&=&48 \end{array}$

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: .$\displaystyle y\:=\:(-4x)^{\frac{1}{2}}$ . . . . 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: .$\displaystyle \begin{array}{cccc}ax+apy &=&ap^2 & [1] \\ bx+bqy &=& bq^2 & [2] \end{array}$

I found the solution of $\displaystyle x$ and $\displaystyle y$ 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?