A tangent with slope m to the parabola $y^2=4x$  which bisect exactly two distinct chords of hyperbola $x^2-4y^2-4=0$  drawn from $(-2,0)$ , then which of the following cannot be true ?

Equation of tangent to parabola is  $y=mx+\frac{1}{m}$
And point R on it is  $(\alpha ,m\alpha +\frac{1}{m})$
R is mid-point of chord AB
$\Rightarrow 2\alpha =h-2,2(m\alpha +\frac{1}{m})=k$ 
$\Rightarrow h=2(\alpha +1),k=2(m\alpha +\frac{1}{m})$  lie on hyperbola
 $4{(\alpha +1)}^2-16{(m\alpha +\frac{1}{m})}^2-4=0$
${\alpha }^2+2\alpha +1-4(m^2{\alpha }^2+\frac{1}{m^2}+2\alpha )-1=0$ 
$(1-4m^2){\alpha }^2-6\alpha -\frac{4}{m^2}=0$ 
$D>0;36+\frac{16}{m^2}(1-4m^2)>0$ 
$\Rightarrow m\in (-\frac{2}{\sqrt{7}},\frac{2}{\sqrt{7}})-\left\{0\right\}$