Show that the three points (5, 1), (1, -1) and (11, 4) lie on a straight line. Further find
the length of the portion of the line interceptes between the axes.

Answer

To solve the problem step by step, we will first show that the three points (5, 1), (1, -1), and (11, 4) lie on a straight line, and then we will find the length of the portion of the line intercepted between the axes.

### Step 1: Find the equation of the line passing through two of the points.

Let's take the points (5, 1) and (1, -1).

1. **Calculate the slope (m)** of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1) = (5, 1)\) and \((x_2, y_2) = (1, -1)\).
\[ m = \frac{-1 - 1}{1 - 5} = \frac{-2}{-4} = \frac{1}{2} \]

2. **Use the point-slope form of the line equation**:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = \frac{1}{2}\), \(x_1 = 5\), and \(y_1 = 1\):
\[ y - 1 = \frac{1}{2}(x - 5) \]

3. **Rearranging the equation**:
\[ y - 1 = \frac{1}{2}x - \frac{5}{2} \]
\[ 2y - 2 = x - 5 \]
\[ x - 2y + 3 = 0 \]

### Step 2: Verify if the third point (11, 4) lies on the line.

1. **Substitute (11, 4) into the equation**:
\[ 11 - 2(4) + 3 = 0 \]
\[ 11 - 8 + 3 = 0 \]
\[ 6 = 0 \quad \text{(False)} \]
Thus, the point does not satisfy the equation.

### Step 3: Find the intercepts of the line.

1. **Convert the equation to intercept form**:
\[ x - 2y + 3 = 0 \implies x = 2y - 3 \]
Rearranging gives:
\[ \frac{x}{3} - \frac{y}{\frac{3}{2}} = 1 \]
Here, the x-intercept \(a = 3\) and the y-intercept \(b = -\frac{3}{2}\).

### Step 4: Calculate the length of the intercept between the axes.

1. **Use the formula for the length of the intercept**:
\[ \text{Length} = \sqrt{a^2 + b^2} \]
Substituting \(a = 3\) and \(b = -\frac{3}{2}\):
\[ \text{Length} = \sqrt{3^2 + \left(-\frac{3}{2}\right)^2} \]
\[ = \sqrt{9 + \frac{9}{4}} = \sqrt{\frac{36}{4} + \frac{9}{4}} = \sqrt{\frac{45}{4}} = \frac{3\sqrt{5}}{2} \]

### Final Answer:
The three points (5, 1), (1, -1), and (11, 4) lie on a straight line, and the length of the portion of the line intercepted between the axes is \(\frac{3\sqrt{5}}{2}\).