The line `3x + 4y -12=0` meets the corrdinates axes at A and B. Find the equation of the circle drawn of AB as diameter.

Answer

To find the equation of the circle drawn with the line segment AB as the diameter, we will follow these steps:

### Step 1: Find the points A and B where the line meets the axes.
The given line is:
\[ 3x + 4y - 12 = 0 \]

**Finding point A (x-intercept):**
To find the x-intercept, set \( y = 0 \):
\[ 3x + 4(0) - 12 = 0 \implies 3x - 12 = 0 \implies 3x = 12 \implies x = 4 \]
Thus, point A is \( (4, 0) \).

**Finding point B (y-intercept):**
To find the y-intercept, set \( x = 0 \):
\[ 3(0) + 4y - 12 = 0 \implies 4y - 12 = 0 \implies 4y = 12 \implies y = 3 \]
Thus, point B is \( (0, 3) \).

### Step 2: Use the coordinates of A and B to find the equation of the circle.
The coordinates of points A and B are:
- A: \( (4, 0) \)
- B: \( (0, 3) \)

The general equation of a circle with diameter endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]
Substituting \( (x_1, y_1) = (4, 0) \) and \( (x_2, y_2) = (0, 3) \):
\[ (x - 4)(x - 0) + (y - 0)(y - 3) = 0 \]

### Step 3: Expand the equation.
Expanding the equation:
\[ (x - 4)(x) + (y)(y - 3) = 0 \]
This simplifies to:
\[ x^2 - 4x + y^2 - 3y = 0 \]

### Step 4: Rearranging the equation.
Rearranging gives us the final equation of the circle:
\[ x^2 + y^2 - 4x - 3y = 0 \]

### Final Answer:
The equation of the circle drawn with AB as diameter is:
\[ x^2 + y^2 - 4x - 3y = 0 \]

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