The point `(2a, a)` lies inside the region bounded by the parabola `x^(2) = 4y` and its latus rectum. Then,

Options

A. `0leale1`
B. `0ltalt1`
C. `agt1`
D. `alt1`

Answer

To solve the problem, we need to determine the conditions under which the point \((2a, a)\) lies inside the region bounded by the parabola \(x^2 = 4y\) and its latus rectum.

### Step 1: Understand the Parabola and its Latus Rectum
The equation of the parabola is given by:
\[ x^2 = 4y \]
This parabola opens upwards. The vertex is at the origin \((0, 0)\).

The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry that passes through the focus. For the parabola \(x^2 = 4y\), the focus is at \((0, 1)\) and the latus rectum is the line \(y = 1\).

### Step 2: Determine the Conditions for the Point
The point \((2a, a)\) must satisfy two conditions:
1. It must lie inside the parabola.
2. Its y-coordinate must be less than 1 (to be below the latus rectum).

### Step 3: Condition for the Point to be Inside the Parabola
To check if the point lies inside the parabola, we substitute \(x = 2a\) and \(y = a\) into the equation of the parabola:
\[ (2a)^2 < 4a \]
This simplifies to:
\[ 4a^2 < 4a \]
Dividing both sides by 4 (assuming \(a \neq 0\)):
\[ a^2 < a \]
Rearranging gives:
\[ a^2 - a < 0 \]
Factoring:
\[ a(a - 1) < 0 \]
This inequality holds when \(0 < a < 1\).

### Step 4: Condition for the Point to be Below the Latus Rectum
The second condition is that the y-coordinate \(a\) must be less than 1:
\[ a < 1 \]
This condition is already satisfied by the first condition.

### Step 5: Conclusion
Combining both conditions, we find that:
\[ 0 < a < 1 \]
Thus, the point \((2a, a)\) lies inside the region bounded by the parabola \(x^2 = 4y\) and its latus rectum when \(a\) is in the interval \( (0, 1) \).

### Final Answer
The point \((2a, a)\) lies inside the region bounded by the parabola \(x^2 = 4y\) and its latus rectum if:
\[ 0 < a < 1 \]