First slide

Definition of a circle

Question

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

Moderate

A

$m \in (1, 2)$
B

$m = 1$
C

$m = 2$
D

$m \in R$

Solution

If the line $y = m x - (m - 1)$ cuts the circle in two distinct points, then
Length of the perpendicular from the centre $<$ Radius.
$\begin{array}{l} \Rightarrow |\frac{m \times 0 - 0 - (m - 1)}{\sqrt{m^{2} + 1}}| < 2 \\ \Rightarrow \frac{| m - 1 |}{\sqrt{m^{2} + 1}} < 2 \\ \Rightarrow (m - 1)^{2} < 4 (m^{2} + 1) \end{array}$
$\Rightarrow 3 m^{2} + 2 m + 3 > 0$
$\Rightarrow 3 \{{(m + \frac{1}{3})}^{2} + \frac{8}{9}\} > 0,,$ which is true for all $m \in R .$
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 $m .$

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