Question

The point $(-2m,m+1)$ is an interior point of the smaller region bounded by the circle $x^2+y^2=4$ and the parabola $y^2=4x$. Then $m$ belongs to the intyerval

• A. $-5-2\sqrt{6}<m<1$
• B. $0<m<4$
• C. $-1<m<\frac{3}{5}$
• D. $-1<m<-5+2\sqrt{6}$

Answer 1

The correct option is D $-1<m<-5+2\sqrt{6}$

When the point $(-2m,m+1)$ lies inside the region bounded by the circle and the parabola
Case - I: Point will lies inside the circle itself
$\Rightarrow S_1<0$
$\Rightarrow (-2m{)}^2+(m+1{)}^2-4<0$
$\Rightarrow 5m^2+2m-3<0$
$\Rightarrow m\in (-1,\frac{3}{5})\cdots \left(i\right)$

Case - II: Point will lies inside the parabola also
$\Rightarrow S_1<0$
$\Rightarrow (m+1{)}^2-4(-2m)<0$
$\Rightarrow m^2+1+10m<0$
$\Rightarrow m\in (-5-2\sqrt{6},-5+2\sqrt{6})\cdots \left(ii\right)$
From equation (i) and (ii),
$-1<m<-5+2\sqrt{6}$