Question

Set of values of $\alpha$ for which the point $(\alpha ,1)$ lies inside the curves $c_1:x^2+y^2-4=0$ and $c_2:y^2=4x$ is

• A. $|\alpha |<\sqrt{3}$
• B. $\alpha >\frac{1}{4}$
• C. $\frac{1}{4}<\alpha <\sqrt{3}$
• D. $\frac{1}{2}<\alpha <\sqrt{2}$

Answer 1

The correct option is C $\frac{1}{4}<\alpha <\sqrt{3}$

$(\alpha ,1)$ lies inside the circle $x^2+y^2-4=0$
$\Rightarrow {\alpha }^2+1-4<0$
$\Rightarrow {\alpha }^2<3,|\alpha |<\sqrt{3}$

$(\alpha ,1)$ lies inside the parabola $y^2=4x$
$\Rightarrow y^2-4x<0$
$\Rightarrow 1-4\alpha <0$
$\Rightarrow \alpha >\frac{1}{4}$
So, $\alpha$ to lie inside both the curves $\frac{1}{4}<\alpha <\sqrt{3}$