Explanation

(c)
Given, equation of hyperbola
$(10x - 5)^{2} + (10y - 4)^{2} = \lambda^{2}(3x + 4y - 1)^{2}$ can be rewritten as
$\frac{\sqrt{\left( x - \frac{1}{2} \right)^{2} + \left( y - \frac{2}{5} \right)^{2}}}{\left| \frac{3x + 4y - 1}{5} \right|} = \left| \frac{\lambda}{2} \right|$
This is of the form of $\frac{PS}{PM} = e$
Where, $P$ is any point on the hyperbola and $S$ is a focus and $M$ is the point of directrix.
Here, $\left| \frac{\lambda}{2} \right| > 1\ \Rightarrow |\lambda| > 2\ \ \ \ (\because e > 1)$
$\Rightarrow \ \ \lambda < - 2$ or $\lambda > 2$