If equation ( 10x - 5 ) ^ 2 + ( 10y - 4 ) ^ 2 = λ ^ 2 ( 3x + 4y - 1...

Given, equation of hyperbola is

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