If the line $y = mx - (m -1)$ cuts the circle $x^2 + y^2 = 4$ at two real and distinct points, then

detailed solution

Correct option is D

If the line y = mx - (m -1) cuts the circle in two distinct points, then     Length of the perpendicular from the centre < Radius.⇒ m×0−0−(m−1)m2+1<2⇒ |m−1|m2+1<2⇒ (m−1)2<4m2+1⇒ 3m2+2m+3>0⇒ 3m+132+89>0,, which is true for all m ∈ R.Hence, option (d) is correct.ALITER The equation of the line is y -1 = m (x -1). Clearly, it passes through (1, 1) which is an interior point of the circle. So, the line cuts the circle for all values of m.