If the point $\left(\lambda^2, \lambda - 2\right)$ is a point lying interior of the region bounded by the parabola $y^2 = 2x$ and the chord joining the point $(2, 2)$ and $(8, -4)$, then the number of the integral values of $\lambda$ is ______.
Step-by-Step Solution
Key Concept: A point lies interior to the region bounded by a parabola and chord if it satisfies two conditions: (1) it lies on the concave side of the parabola, meaning y² < 2x for point (λ², λ-2), which gives (λ-2)² < 2λ²; and (2) it lies on the same side of the chord as the interior, using the linear inequality from the chord equation x + y - 4 = 0.
For point $P(\lambda^2, \lambda-2)$ to lie inside the parabola $y^2 = 2x$, we need $(\lambda-2)^2 0$ with roots $\lambda = -2 \pm 2\sqrt{2}$. The chord $AB$ has equation $y - 2 = \frac{-4-2}{8-2}(x-2) = x + y - 4 = 0$. For $P$ and $O$ to lie on the same side of this line, we need $\lambda^2 + \lambda - 6 < 0$, which factors as $(\lambda+3)(\lambda-2) < 0$ giving $-3 < \lambda < 2$. Combining both conditions yields $-2 + 2\sqrt{2} < \lambda < 2$.
Correct Answer: 1