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

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$m = - \frac{1}{2}$

$m = \frac{1}{2}$

$m = (\frac{2}{\sqrt{7}} - 1)$

$m = 0$

answer is D.

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Detailed Solution

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