Find the calculation whether the points (13, 8), (26, -4) lie in the same, adjacent, or opposite angles formed by the straight lines `5x+6y-112=0, and 10x+11y-217=0`.

Answer

To determine whether the points (13, 8) and (26, -4) lie in the same, adjacent, or opposite angles formed by the lines \(5x + 6y - 112 = 0\) and \(10x + 11y - 217 = 0\), we will follow these steps:

### Step 1: Substitute the points into the first line equation
We will first substitute the points into the equation of the first line \(5x + 6y - 112 = 0\).

1. **For point (13, 8)**:
\[ 5(13) + 6(8) - 112 = 65 + 48 - 112 = 1 \]
The result is \(1\), which is greater than \(0\).

2. **For point (26, -4)**:
\[ 5(26) + 6(-4) - 112 = 130 - 24 - 112 = -6 \]
The result is \(-6\), which is less than \(0\).

### Step 2: Analyze the results for the first line
Since the point (13, 8) gives a positive result and (26, -4) gives a negative result, the points lie on opposite sides of the line \(5x + 6y - 112 = 0\).

### Step 3: Substitute the points into the second line equation
Next, we will substitute the points into the equation of the second line \(10x + 11y - 217 = 0\).

1. **For point (13, 8)**:
\[ 10(13) + 11(8) - 217 = 130 + 88 - 217 = 1 \]
The result is \(1\), which is greater than \(0\).

2. **For point (26, -4)**:
\[ 10(26) + 11(-4) - 217 = 260 - 44 - 217 = -1 \]
The result is \(-1\), which is less than \(0\).

### Step 4: Analyze the results for the second line
Similar to the first line, the point (13, 8) gives a positive result and (26, -4) gives a negative result. Thus, the points also lie on opposite sides of the line \(10x + 11y - 217 = 0\).

### Conclusion
Since both points (13, 8) and (26, -4) lie on opposite sides of both lines, they form opposite angles with respect to the two lines.

### Final Answer
The points (13, 8) and (26, -4) lie in opposite angles formed by the lines \(5x + 6y - 112 = 0\) and \(10x + 11y - 217 = 0\).

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