The number of integral point inside the triangle made by the line ` 3x + 4y - 12 =0` with the coordinate axes which are equidistant from at least two sides is/are :
( an integral point is a point both of whose coordinates are integers. )

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem of finding the number of integral points inside the triangle formed by the line \(3x + 4y - 12 = 0\) with the coordinate axes that are equidistant from at least two sides, we can follow these steps:

### Step 1: Find the intercepts of the line with the axes
The line equation can be rewritten in intercept form:
\[ 3x + 4y = 12 \]
To find the x-intercept, set \(y = 0\):
\[ 3x = 12 \implies x = 4 \]
So, the x-intercept is \((4, 0)\).

To find the y-intercept, set \(x = 0\):
\[ 4y = 12 \implies y = 3 \]
So, the y-intercept is \((0, 3)\).

### Step 2: Identify the vertices of the triangle
The triangle is formed by the intercepts and the origin:
- Vertex A: \((0, 0)\)
- Vertex B: \((4, 0)\)
- Vertex C: \((0, 3)\)

### Step 3: Determine the area of the triangle
The area \(A\) of the triangle formed by the vertices \((0, 0)\), \((4, 0)\), and \((0, 3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 3 = 6 \]

### Step 4: Find the integral points inside the triangle
We are looking for integral points \((h, k)\) such that they are inside the triangle and equidistant from at least two sides. The condition for a point to be equidistant from the x-axis and y-axis is when \(h = k\).

### Step 5: Set up inequalities for the points inside the triangle
The point \((h, k)\) must satisfy:
1. \(h \geq 0\)
2. \(k \geq 0\)
3. The line equation must be satisfied: \(3h + 4k < 12\)

Substituting \(k = h\) into the line inequality gives:
\[ 3h + 4h < 12 \implies 7h < 12 \implies h < \frac{12}{7} \approx 1.71 \]

### Step 6: Determine integral values for \(h\) and \(k\)
Since \(h\) must be a non-negative integer and less than \(1.71\), the only possible integral value for \(h\) is:
- \(h = 1\)

Thus, if \(h = 1\), then \(k = 1\).

### Step 7: Conclusion
The only integral point inside the triangle that is equidistant from at least two sides is \((1, 1)\).

### Final Answer
The number of integral points inside the triangle that are equidistant from at least two sides is **1**.