If the line ax + y = c , touches both the curves x ^ 2 + y ^ 2 = 1...

The equation of the tangent to the parabola ${y}^{2} = 4ax$ is $y = mx + \dfrac{a}{m} .$

Hence, the equation of tangent to the parabola ${y}^{2} = 4\sqrt{2}x$ is $y = mx + \dfrac{\sqrt{2}}{m} . . . \left(1\right)$

A line $y = mx + c$ is a tangent to the circle ${x}^{2} + {y}^{2} = {r}^{2}$ if ${c}^{2} = {r}^{2}\left(1 + {m}^{2}\right) .$

Hence, the line \left(1\right)$is also a tangent to the circle${x}^{2} + {y}^{2} = 1,$if${\left(\dfrac{\sqrt{2}}{m}\right)}^{2} = 1\left({m}^{2} + 1\right)$</p><p>$⇒ {m}^{2}\left({m}^{2} + 1\right) - 2 = 0$</p><p>$⇒ {m}^{4} + {m}^{2} - 2 = 0$</p><p>$⇒ \left({m}^{2} + 2\right)\left({m}^{2} - 1\right) = 0$</p><p>$⇒ {m}^{2} = 1$since${m}^{2} + 2 \ne 0$</p><p>$⇒ m = \pm 1 .$</p><p>Thus, the equation of the common tangent is$y = \pm x \pm \sqrt{2}, ⇒ \pm x - y = ∓ \sqrt{2} .$</p><p>Given, the equation of the common tangent is$ax + y = c$, hence,$c = \pm \sqrt{2}$and \left|c\right| = \sqrt{2} .$