The equation of a line is `3x - 4y + 12 = 0`. It meets the x-axis at point A and the y-axis at point B. Find :
the length of intercept AB, cut by the line within the co-ordinate axes.

Answer

To find the length of intercept AB cut by the line within the coordinate axes, we will follow these steps:

### Step 1: Identify the equation of the line
The given equation of the line is:
\[ 3x - 4y + 12 = 0 \]

### Step 2: Find the x-intercept (Point A)
To find the x-intercept, we set \( y = 0 \) in the equation of the line:
\[ 3x - 4(0) + 12 = 0 \implies 3x + 12 = 0 \implies 3x = -12 \implies x = -4 \]
Thus, the coordinates of point A (x-intercept) are:
\[ A(-4, 0) \]

### Step 3: Find the y-intercept (Point B)
To find the y-intercept, we set \( x = 0 \) in the equation of the line:
\[ 3(0) - 4y + 12 = 0 \implies -4y + 12 = 0 \implies -4y = -12 \implies y = 3 \]
Thus, the coordinates of point B (y-intercept) are:
\[ B(0, 3) \]

### Step 4: Calculate the length of intercept AB
The length of the line segment AB can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
\[ AB = \sqrt{(0 - (-4))^2 + (3 - 0)^2} = \sqrt{(0 + 4)^2 + (3)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Final Answer
The length of intercept AB is:
\[ AB = 5 \]

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