Quadratic EquationHard
Question
The number of real roots of the equation $\sqrt{|1 - x|} = kx$, where $k$ is a parameter is
Options
A.1 , if $k > \frac{1}{2}$
B.2 , if $k = \frac{1}{2}$
C.3, if $0 < k < \frac{1}{2}$
D.1 , if $k \leq 0$
Solution
$\sqrt{|1 - x|} = kx$
For tangency of two curves
$$\begin{matrix} & \sqrt{x_{1} - 1} = {kx}_{1} \\ \text{~and~} & \frac{1}{2\sqrt{x_{1} - 1}} = k \\ \Rightarrow & \sqrt{x_{1} - 1} = \frac{x_{1}}{2\sqrt{x_{1} - 1}} \\ \Rightarrow & 2\left( x_{1} - 1 \right) = x_{1} \Rightarrow x_{1} = 2,k = \frac{1}{2} \end{matrix}$$
From graph it is clear that equation has
$$\begin{matrix} \text{~one solution if~}k & \ \in ( - \infty,0\rbrack \cup \left( \frac{1}{2},\infty \right) \\ \text{~two solutions if~}k & \ = \frac{1}{2} \\ \text{~three solutions if~}k & \ \in \left( 0,\frac{1}{2} \right) \end{matrix}$$
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