Oxford University MAT 2016 Solved Papers

Let $\mathit{a}\mathit{x}+\mathit{b}\mathit{y}=\mathit{c}$, which rearranges to $\mathit{y}=\frac{\mathit{a}}{\mathit{b}}\mathit{x}+\frac{\mathit{c}}{\mathit{b}}$. Given that $\mathit{b}$ is positive we can interpret this as achieving the maximum $\mathit{c}$ when the line $\mathit{y}=\frac{\mathit{a}}{\mathit{b}}\mathit{x}+\frac{\mathit{c}}{\mathit{b}}$ is moved up the $\mathit{y}$-axis whilst still intersecting the disc formed by ${\mathit{x}}^2+{\mathit{y}}^2\le 1$. Hence the line should be tangent to the unit circle.
By Pythagoras,
${\left(\frac{\mathit{c}}{\mathit{b}}\right)}^2=1+{\left(\frac{\mathit{a}}{\mathit{b}}\right)}^2$