The circle that can be drawn to touch the coordinate axes and the line `4x + 3y =12` cannot lie in

Options

A. first quadrant
B. second quadrant
C. third quadrant
D. fourth quadrant

Answer

To solve the problem, we need to determine in which quadrant a circle that touches the coordinate axes and the line \(4x + 3y = 12\) cannot lie.

### Step-by-Step Solution:

1. **Identify the Line Intercepts**:
- To find where the line \(4x + 3y = 12\) intersects the axes, we can find the x-intercept and y-intercept.
- **X-intercept**: Set \(y = 0\):
\[ 4x + 3(0) = 12 \implies 4x = 12 \implies x = 3 \]
So, the x-intercept is \((3, 0)\).
- **Y-intercept**: Set \(x = 0\):
\[ 4(0) + 3y = 12 \implies 3y = 12 \implies y = 4 \]
So, the y-intercept is \((0, 4)\).

2. **Draw the Axes and Line**:
- We can visualize the coordinate axes and the line. The line passes through the points \((3, 0)\) and \((0, 4)\).

3. **Determine the Triangle Formed**:
- The line and the axes form a right triangle in the first quadrant with vertices at \((0, 0)\), \((3, 0)\), and \((0, 4)\).

4. **Circle Touching the Axes**:
- A circle that touches both axes must have its center at \((r, r)\) where \(r\) is the radius of the circle. This circle will touch the x-axis and y-axis at points \((r, 0)\) and \((0, r)\) respectively.

5. **Incenter of the Triangle**:
- The incenter of the triangle formed by the axes and the line is the point where the circle that touches all three sides of the triangle will be located. The incenter will be inside the triangle.

6. **Quadrants Analysis**:
- The circle that touches the coordinate axes and the line \(4x + 3y = 12\) can only lie in the first quadrant where both x and y are positive.
- It cannot lie in the third quadrant because in the third quadrant both x and y are negative, and a circle cannot touch the positive axes there.

### Conclusion:
The circle that can be drawn to touch the coordinate axes and the line \(4x + 3y = 12\) cannot lie in the **third quadrant**.