Question

The range of values of $m$ for which the line $y=mx+2$ cuts the circle $x^2+y^2=1$ at distinct and coincident points is

• A. $(-\infty ,-\sqrt{3}]\cup [\sqrt{3},+\infty )$
• B. $[-\sqrt{3},\sqrt{3}]$
• C. $[\sqrt{3},+\infty )$
• D. none of these

Answer 1

The correct option is A $(-\infty ,-\sqrt{3}]\cup [\sqrt{3},+\infty )$

For the given condition, distance of the line from the centre should be less than or equal to radius of the circle.
$\Rightarrow \left|\frac{2}{\sqrt{1+m^2}}\right|\le 1$
$\Rightarrow m^2+1\ge 4\Rightarrow m^2\ge 3\to m\in (-\infty ,-\sqrt{3}]\cup [\sqrt{3},\infty )$