Question

If the point (λ, λ + 1) lies inside the region bounded by the curve $x=\sqrt{25-y^2}$ and y-axis, then λ belongs to the interval
(a) (−1, 3)
(b) (−4, 3)
(c) (−∞, −4) ∪ (3, ∞)
(d) none of these

Answer 1

(a) (−1, 3)

The given equation of the curve is $x^2+y^2=25$.

Since (λ, λ + 1) lies inside the region bounded by the curve $x^2+y^2=25$ and the y-axis, we have:

${\lambda }^2+{\left(\lambda +1\right)}^2<25$, $\mathrm{provided}\lambda +1>0$

$$\begin{gathered} \Rightarrow {\lambda }^2+{\lambda }^2+1+2\lambda <25,\lambda >-1 \\ \Rightarrow 2{\lambda }^2+2\lambda -24<0,\lambda >-1 \\ \Rightarrow {\lambda }^2+\lambda -12<0,\lambda >-1 \\ \Rightarrow \left(\lambda -3\right)\left(\lambda +4\right)<0,\lambda >-1 \\ \Rightarrow -4<\lambda <3,\lambda >-1 \\ \Rightarrow \lambda \in \left(-1,3\right) \end{gathered}$$