If three normals are drawn from the point& (c, 0) to the parabola~...

Equation of the normal in slope form is $y = mx - 2m - m^{3}$ which passes through \left(c, 0\right)$<br />$⇒ 0 = mc - 2m - m^{3}$<br />$⇒ m^{3} + \left(2 - c\right)m = 0$<br />$⇒ m = 0~or&m^{2} + \left(2 - c\right) = 0$<br />$⇒ {m}_{1}{m}_{2} = \dfrac{2 - c}{1} = - 1$<br />$⇒ c = 3