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

Options

• (−1, 3)

• (−4, 3)

• (−∞, −4) ∪ (3, ∞)

• none of these

Answer 1

(−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:

\[\lambda^2 + \left( \lambda + 1 \right)^2 < 25\],

\[\text { provided } \lambda + 1 > 0\]

\[\Rightarrow \lambda^2 + \lambda^2 + 1 + 2\lambda < 25, \lambda > - 1\]

\[ \Rightarrow 2 \lambda^2 + 2\lambda - 24 < 0, \lambda > - 1\]

\[ \Rightarrow \lambda^2 + \lambda - 12 < 0, \lambda > - 1\]

\[ \Rightarrow \left( \lambda - 3 \right)\left( \lambda + 4 \right) < 0, \lambda > - 1\]

\[ \Rightarrow - 4 < \lambda < 3, \lambda > - 1\]

\[ \Rightarrow \lambda \in \left( - 1, 3 \right)\]