Question

If the point $(\lambda ,\lambda +1)$ lies inside the region bounded by the curve $x=\sqrt{25-y^2}$ and y-axis, then $\lambda$ belongs to the interval

• A. $(-1,3)$
• B. $(-4,3)$
  
• C. $(-\infty ,-4)\cup (3,\infty )$
  
• D. none of these

Answer 1

The correct option is A $(-1,3)$

The given equation of the curve is $x=\sqrt{25-y^2}$
Squaring on both sides, we get
$\Rightarrow x^2=25-y^2$
$\Rightarrow x^2+y^2=25$
Since $(\lambda ,\lambda +1)$ lies inside the region bounded by the curve $x^2+y^2=25$ and the y-axis,

we have: ${\lambda }^2+(\lambda +1{)}^2<25$, provided $\lambda +1>0$

$\Rightarrow {\lambda }^2+{\lambda }^2+1+2\lambda <25,\lambda >-1$ [Since, $(a+b{)}^2=a^2+2ab+b^2$]

$\Rightarrow 2{\lambda }^2+2\lambda -24<0,\lambda >-1$

$\Rightarrow {\lambda }^2+\lambda -12<0,\lambda >-1$

$\Rightarrow (\lambda -3)(\lambda +4)<0,\lambda >-1$

$\Rightarrow -4<\lambda <3,\lambda >-1$

$\Rightarrow \lambda \in (-4,3)$ and $\lambda \in (-1,\infty )$

$\therefore \lambda \in (-1,3)$