First slide

Analysis of two circles in circles

Question

The range of m for which the line $y = m x + 2$ cuts the circle $x^{2} + y^{2} = 1$ at distinct or coincident point, is

Difficult

A

$[- \sqrt{3}, \sqrt{3}]$
B

$(- \infty, - \sqrt{3}] \cup [\sqrt{3}, + \infty)$
C

$[\sqrt{3}, + \infty)$
D

None of these

Solution

The length of the perpendicular from the centre $(0, 0)$ to the line $= \frac{2}{\sqrt{(1 + m^{2})}} .$
The radius of the circle $= 1$
For the line to cut the circle at distinct or coincident points, $\frac{2}{\sqrt{{(1 + m)}^{2}}} \leq 1 \Rightarrow 1 + m^{2} \geq 4 \Rightarrow m^{2} \geq 3.$
$∴ m \in (- \infty, - \sqrt{3}] \cup [\sqrt{3}, + \infty)$

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