Question

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. $0<\alpha <\frac{1}{2}$

Answer 1

Answer: (A)

$\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}$

Given Lines

$y-\sqrt{1-x^2}=0$ and $x+y-1=0$

Given point $(\alpha ,\alpha )$ lies between both lines hence

By position of point w.r to line concept

$\alpha -\sqrt{1-{\alpha }^2}<0$

$\alpha <\sqrt{1-{\alpha }^2}$

On squaring

${\alpha }^2<1-{\alpha }^2$

$2{\alpha }^2<1$

$\alpha <\frac{1}{\sqrt{2}}$

Similarly $\alpha +\alpha -1>0$

$2\alpha >1$

$\alpha >\frac{1}{2}$

Hence $\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}$