Solve $(a-b)^2$ with $(r_1a-r_2b)^2$

Question

Is it possible to solve for $$(a-b)^2$$

by knowing only the following information: $$(r_1a-r_2b)^2 = C$$ $$r_1$$ $$r_2$$ $$C$$

Answer 1

Let $x=a-b$. So we are trying to find $x^2$.

We know that $(r_1a-r_2b)^2=C$. Next substitute in for $x$:

$$(r_1(x+b)-r_2b)^2=C$$

$$r_1(x+b)-r_2b=\pm\sqrt{C}$$

$$x+b-\frac{r_2b}{r_1}=\pm\frac{\sqrt{C}}{r_1}$$

$$x=-b+\frac{r_2b}{r_1}\pm\frac{\sqrt{C}}{r_1}$$

$$x^2=\left(-b+\frac{r_2b}{r_1}\pm\frac{\sqrt{C}}{r_1}\right)^2$$

Note that $x^2$ depends upon the value of $b$ which we do not know so it can not be found from the information given.