Fill in the blanks
(e) Angle between lines `5x + y = 7` and `–x + 5y = 9` is _______.

Answer

To find the angle between the lines given by the equations \(5x + y = 7\) and \(-x + 5y = 9\), we will follow these steps:

### Step 1: Identify the slopes of the lines
The general form of a line is given by \(Ax + By + C = 0\). The slope \(m\) of the line can be calculated using the formula:
\[ m = -\frac{A}{B} \]

#### For the first line \(5x + y = 7\):
Rearranging it to the standard form:
\[ 5x + y - 7 = 0 \]
Here, \(A = 5\) and \(B = 1\). Therefore, the slope \(m_1\) is:
\[ m_1 = -\frac{5}{1} = -5 \]

#### For the second line \(-x + 5y = 9\):
Rearranging it to the standard form:
\[ -x + 5y - 9 = 0 \]
Here, \(A = -1\) and \(B = 5\). Therefore, the slope \(m_2\) is:
\[ m_2 = -\frac{-1}{5} = \frac{1}{5} \]

### Step 2: Use the formula for the angle between two lines
The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be calculated using the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the slopes into the formula
Substituting \(m_1 = -5\) and \(m_2 = \frac{1}{5}\):
\[ \tan \theta = \left| \frac{-5 - \frac{1}{5}}{1 + (-5) \cdot \frac{1}{5}} \right| \]
Calculating the numerator:
\[ -5 - \frac{1}{5} = -\frac{25}{5} - \frac{1}{5} = -\frac{26}{5} \]
Calculating the denominator:
\[ 1 + (-5) \cdot \frac{1}{5} = 1 - 1 = 0 \]

### Step 4: Analyze the result
Since the denominator is 0, \(\tan \theta\) becomes undefined, which indicates that:
\[ \theta = \frac{\pi}{2} \text{ or } 90^\circ \]

### Final Answer
Thus, the angle between the lines \(5x + y = 7\) and \(-x + 5y = 9\) is:
\[ \text{Angle} = 90^\circ \]

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