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

Options

A. `m in (1, 2)`
B. `m=1`
C. `m=2`
D. `m in R`

Answer

D

Explanation

If the line `y= mx-(m-1)` cuts the circle in two distinct points, then
Length of the perpendicular from the centre `lt` Radius.
`rArr |(mxx0-0-(m-1))/(sqrt(m^(2)+1))| lt 2`
`rArr (|m-1|)/(sqrt(m^(2)+1))lt2`
`rArr (m-1)^(2)lt 2(m^(2)+1)`
`rArr m^(2)+2m+1 gt 0`
`rArr (m+1)^(2)gt0`, which is true for all `m in R`.
Hence, option (d) is correct.
`ul("ALITER")` The equation of the line is `y-1=m(-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.