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) \[\frac{1}{2}<\alpha <\frac{1}{3}\]

C) \[\frac{1}{3}<\alpha <\frac{1}{\sqrt{3}}\]

D) \[\frac{1}{4}<\alpha <\frac{1}{2}\]

Correct Answer: C

Solution :

The point should lies on the opposite side of the origin of the line\[~x+y-1=0\] Then,    \[\alpha +\alpha -1>0\] \[\Rightarrow \]               \[2\alpha \ge 1\Rightarrow \alpha >\frac{1}{2}\]                               ... (i) Also,      \[({{\alpha }^{2}}+{{\alpha }^{2}})<1\] \[\Rightarrow \]               \[\left( \frac{-1}{\sqrt{2}} \right)<\alpha <\left( \frac{1}{\sqrt{2}} \right)\]                            ? (ii) From Eqs. (i) and (ii), we get                 \[\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}\]