question_answer

The range of \['\alpha '\] for which the point, \[(\alpha ,\,\,\alpha )\] lies inside the region bounded by the curves \[y=\sqrt{1-{{x}^{2}}}\]  and \[x+y=1\] is

A) \[\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}\]                     

B) \[-\frac{1}{\sqrt{2}}<\alpha <\frac{1}{\sqrt{2}}\]

C) \[\alpha >\frac{1}{2}\]              

D) \[0<\alpha <\frac{1}{2}\]

Correct Answer: A

Solution :

Point\[(\alpha ,\,\,\alpha )\] will lie is shaded region
it          \[{{a}^{2}}+{{a}^{2}}-1<0\]
\[\Rightarrow \]   \[2{{a}^{2}}-1<0\]
\[\Rightarrow \]   \[-\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}\]	?(1)
and       \[\alpha +\alpha >1\]            \[\Rightarrow \] \[\alpha >\frac{1}{2}\]	?(2)
\[\therefore \]      common solution of (1) and (2) is:
\[\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}\]