where did the $a,b,c$ are positive real numbers come from? I don't see $a,b,c$ in the problem. ANyway, it's a trivial case of Cauchy-Schwarz. We consider the discriminant of $(a_1x-b_1)^2+(a_2x-b_2)^2+..+(a_nx-b_n)^2 = 0$. (THis is pretty well-known)

Who needs the discriminant? with $\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n} = x$, we have $\sum_{i=1}^{n} a^{2}_i \cdot \sum_{i=1}^{n} b^{2}_i =x^2 \left (\sum_{i=1}^{n} b^{2}_i\right )^2 = \left (\sum_{i=1}^{n} a_{i}b_{i}\right )^2$. Strange, asking this question ...