Fitting an inverse quadratic curve

[likwid4]

If I know 2 points in the +x,+y quadrant, and I want to find the

f(x) =    a
         --------
         (b*x)^2

curve that passes through both points (a and b are constants).

This is probably either really simple or impossible.

 

[Moo Of Doom]

Yes... this is either really simple or impossible.

Let $$k=\frac{a}{b^2}$$. Since $$f(x) = \frac{a}{(bx)^2} = \frac{a}{b^2x^2}=\frac{\frac{a}{b^2}}{x^2}$$, we can say $$f(x) = \frac{k}{x^2}$$.

If the two points are given by $$P_1 = (x_1,y_1)$$ and $$P_2 = (x_2,y_2)$$, then they both lie on such a curve if and only if $$y_1 = \frac{k}{x_1^2}$$ and $$y_2 = \frac{k}{x_2^2}$$. That is, if $$x_1^2y_1=x_2^2y_2$$. If this is the case, then let $$k=x_1^2y_1$$, and translate to a and b. Since we've got two variables and one equation, we're free to choose one of them. For ease, let's make b=1. Then a=k. So, $$f(x)=\frac{k}{x^2}$$.

 

[likwid4]

Though I can't think how I missed that but I do feel quite stupid now