If y = m _ 1 x + c _ 1 and y = m _ 2 x + c _ 2 , m _ 1 ≠ m _ 2 are...

Equation of the tangent to the parabola can be written as $y = mx + \left(\dfrac{1}{4m}\right)$ or $y - mx - \dfrac{1}{4m} = 0$

Now, the perpendicular distance from \left(0, 0\right)$to the tangent is equal to the radius of the given circle, so$OM = \sqrt{2} ⇒ \left|\dfrac{\dfrac{- 1}{4m}}{\sqrt{1 + {m}^{2}}}\right| = \sqrt{2}$</p><p>$⇒ 2\left(16{m}^{2}\right)\left(1 + {m}^{2}\right) = 1 ⇒ 32{m}^{4} + 32{m}^{2} - 1 = 0$</p><p>$⇒ {m}^{2} = \dfrac{- {2}^{5} + {2}^{3}\sqrt{16 + 2}}{2 \times {2}^{5}}$ignoring the negative sign as it is square function.</p><p>Now \left|8{m}_{1}{m}_{2}\right| = \left|8{m}^{2}\right|$

= \left|\dfrac{- {2}^{5} + {2}^{3}\sqrt{16 + 2}}{{2}^{3}}\right| = 3\sqrt{2} - 4