Question

If the line $y-1=m(x-1)$ cuts the circle $x^2+y^2=4$ at two real points then the number of possible values of $m$ is:

• A. $1$
• B. $2$
• C. Infinite
• D. None of these

Answer 1

Answer: (B)

$2$

Given circle is $x^2+y^2=4$,

Given that the line $y-1=m(x-1)$ intersects the circle at two different points

If the perpendicular distance from the centre of the circle to the line is less than the radius of the circle then the line intersects at two different real points

$⟹\frac{|m-1|}{\sqrt{1+m^2}}<2$

Squaring on both sides gives,

$m^2-2m+1<4+4m^2⟹3m^2+2m+3>0$,

Given quadratic equation has complex roots and the co-efficient of $x^2$ is positive

$\therefore$ The quadratic equation is always positive

Hence, infinite values of $m$ exist to intersect the line at $2$ diffferent real points.