If `(9a, 6a)` is a point bounded in the region formed by parabola `y^(2)=16x and x=9`, then

Options

A. `a in (0, 1)`
B. `alt(1)/(4)`
C. `a lt 1`
D. `0lt alt4`

Answer

To solve the problem, we need to determine the values of \( a \) such that the point \( (9a, 6a) \) lies within the region bounded by the parabola \( y^2 = 16x \) and the line \( x = 9 \).

### Step 1: Analyze the parabola and line
The equation of the parabola is given by \( y^2 = 16x \). This can be rewritten as:
\[ x = \frac{y^2}{16} \]
The line \( x = 9 \) is a vertical line. The region bounded by the parabola and the line is to the left of the line \( x = 9 \).

### Step 2: Determine the conditions for the point \( (9a, 6a) \)
For the point \( (9a, 6a) \) to lie inside the region formed by the parabola and the line, it must satisfy two conditions:
1. It must be below the parabola: \( y^2 < 16x \)
2. It must be to the left of the line: \( x < 9 \)

### Step 3: Apply the conditions
**Condition 1: Below the parabola**
Substituting \( x = 9a \) and \( y = 6a \) into the parabola's equation:
\[ (6a)^2 < 16(9a) \]
This simplifies to:
\[ 36a^2 < 144a \]
Rearranging gives:
\[ 36a^2 - 144a < 0 \]
Factoring out \( 36a \):
\[ 36a(a - 4) < 0 \]
This inequality holds when \( a \) is between the roots:
\[ 0 < a < 4 \]

**Condition 2: To the left of the line**
For the point to be to the left of the line \( x = 9 \):
\[ 9a < 9 \]
Dividing both sides by 9 gives:
\[ a < 1 \]

### Step 4: Combine the conditions
From Condition 1, we have \( 0 < a < 4 \).
From Condition 2, we have \( a < 1 \).

The common values for \( a \) that satisfy both conditions are:
\[ 0 < a < 1 \]

### Conclusion
Thus, the point \( (9a, 6a) \) is bounded in the region formed by the parabola \( y^2 = 16x \) and the line \( x = 9 \) when:
\[ a \in (0, 1) \]