If the point ( , +1) lies inside the region bounded by the curve … - Vidyadip

MCQ

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

✓
$(-1,\ 3)$
B
$(-4,\ 3)$
C
$(-\infty,\ -4)\cup(3,\ \infty)$
D
None of these

✓

Answer

Correct option: A.

$(-1,\ 3)$

$(-1,\ 3)$

Solution:
The given equation of the curve is $x^2 + y^2 = 25$
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 $\lambda+1>0$
$\Rightarrow\lambda^2+\lambda^2+12\lambda<25,\ \lambda>-1$
$\Rightarrow2\lambda^2+2\lambda-24<0,\ \lambda>-1$
$\Rightarrow\lambda^2+\lambda-12<0,\ \lambda>-1$
$\Rightarrow(\lambda-3)(\lambda+4)<0,\ \lambda>-1$
$\Rightarrow-4<\lambda<3,\ \lambda>-1$
$\Rightarrow\lambda\in(-1,\ 3)$

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