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. $|\alpha |<2$
• C. $\frac{1}{4}<\alpha <\sqrt{3}$
• D. None of these

Answer 1

Answer: (C)

$\frac{1}{4}<\alpha <\sqrt{3}$

For $(\alpha ,1)$ to lie inside the curve $C_1$,
${\alpha }^2+1-4<0$
${\alpha }^2<3$
$\Rightarrow \alpha <\sqrt{3}$
For $(\alpha ,1)$ to lie inside the curve $C_2$,
$1-4\alpha <0$
$\Rightarrow \frac{1}{4}<\alpha$
Hence, $\frac{1}{4}<\alpha <\sqrt{3}$