Point `(0, beta)` lies on or inside the triangle fromed by the lines ` y=0, x+y=8 ` and ` 3x - 4y + 12 =0`. Then ` beta` can be :

Options

A. 2
B. 4
C. 8
D. 12

Answer

To solve the problem, we need to determine the range of the value of \( \beta \) for the point \( (0, \beta) \) that lies on or inside the triangle formed by the lines \( y = 0 \), \( x + y = 8 \), and \( 3x - 4y + 12 = 0 \).

### Step 1: Identify the lines and their intersections
1. The first line is \( y = 0 \), which is the x-axis.
2. The second line is \( x + y = 8 \). We can find its intercepts:
- When \( x = 0 \), \( y = 8 \) (y-intercept).
- When \( y = 0 \), \( x = 8 \) (x-intercept).
- Thus, the line intersects the axes at points \( (0, 8) \) and \( (8, 0) \).

3. The third line is \( 3x - 4y + 12 = 0 \). Rearranging gives:
\[ 4y = 3x + 12 \implies y = \frac{3}{4}x + 3 \]
- The y-intercept is \( (0, 3) \).
- To find the x-intercept, set \( y = 0 \):
\[ 0 = \frac{3}{4}x + 3 \implies \frac{3}{4}x = -3 \implies x = -4 \]
- Thus, the line intersects the axes at points \( (0, 3) \) and \( (-4, 0) \).

### Step 2: Find the vertices of the triangle
Now we need to find the intersection points of the lines to determine the vertices of the triangle:
1. Intersection of \( x + y = 8 \) and \( 3x - 4y + 12 = 0 \):
- From \( x + y = 8 \), we have \( y = 8 - x \).
- Substitute into \( 3x - 4(8 - x) + 12 = 0 \):
\[ 3x - 32 + 4x + 12 = 0 \implies 7x - 20 = 0 \implies x = \frac{20}{7} \]
- Substitute back to find \( y \):
\[ y = 8 - \frac{20}{7} = \frac{56 - 20}{7} = \frac{36}{7} \]
- Thus, one vertex is \( \left(\frac{20}{7}, \frac{36}{7}\right) \).

2. Intersection of \( y = 0 \) and \( x + y = 8 \):
- Set \( y = 0 \) in \( x + y = 8 \):
\[ x = 8 \implies (8, 0) \]

3. Intersection of \( y = 0 \) and \( 3x - 4y + 12 = 0 \):
- Set \( y = 0 \) in \( 3x + 12 = 0 \):
\[ 3x = -12 \implies x = -4 \implies (-4, 0) \]

### Step 3: Determine the area of the triangle
The vertices of the triangle are \( (8, 0) \), \( (-4, 0) \), and \( \left(\frac{20}{7}, \frac{36}{7}\right) \).

### Step 4: Determine the range of \( \beta \)
For the point \( (0, \beta) \) to lie inside or on the boundary of the triangle:
- The y-coordinate \( \beta \) must be between the y-values of the triangle vertices.
- The maximum y-value occurs at \( (0, 3) \) and the minimum at \( (0, 0) \).

Thus, the range of \( \beta \) is:
\[ 0 \leq \beta \leq 3 \]

### Conclusion
The possible values of \( \beta \) are from \( 0 \) to \( 3 \).