To find the value of \( a \) such that the line \( 2y - ax + 4 = 0 \) is perpendicular to the line \( y + \frac{x}{4} - 7 = 0 \), we need to follow these steps: ### Step 1: Determine the slope of the second line The equation of the second line is given as: \[ y + \frac{x}{4} - 7 = 0 \] We can rearrange this equation into the slope-intercept form \( y = mx + b \), where \( m \) is the slope: \[ y = -\frac{1}{4}x + 7 \] From this, we can see that the slope \( m_2 \) of the second line is: \[ m_2 = -\frac{1}{4} \] ### Step 2: Determine the slope of the first line Next, we need to rearrange the first line \( 2y - ax + 4 = 0 \) into the slope-intercept form. We can do this by isolating \( y \): \[ 2y = ax - 4 \] \[ y = \frac{a}{2}x - 2 \] From this, we can see that the slope \( m_1 \) of the first line is: \[ m_1 = \frac{a}{2} \] ### Step 3: Use the condition for perpendicular lines For two lines to be perpendicular, the product of their slopes must equal \(-1\): \[ m_1 \cdot m_2 = -1 \] Substituting the slopes we found: \[ \left(\frac{a}{2}\right) \cdot \left(-\frac{1}{4}\right) = -1 \] ### Step 4: Solve for \( a \) Now we can solve the equation: \[ -\frac{a}{8} = -1 \] To eliminate the negative signs, we can multiply both sides by \(-1\): \[ \frac{a}{8} = 1 \] Next, multiply both sides by \( 8 \): \[ a = 8 \] ### Conclusion Thus, the value of \( a \) is \( 8 \). Therefore, the correct option is: **C. 8** ### Explanation of Other Options - **A. -4**: This value does not satisfy the condition for perpendicular slopes. If \( a = -4 \), then \( m_1 = -2 \) and \( m_1 \cdot m_2 = -2 \cdot -\frac{1}{4} = \frac{1}{2} \), which is not equal to \(-1\). - **B. 4**: If \( a = 4 \), then \( m_1 = 2 \) and \( m_1 \cdot m_2 = 2 \cdot -\frac{1}{4} = -\frac{1}{2} \), which is also not equal to \(-1\). - **D. -8**: If \( a = -8 \), then \( m_1 = -4 \) and \( m_1 \cdot m_2 = -4 \cdot -\frac{1}{4} = 1 \), which again does not satisfy the perpendicular condition. ### Revision Summary - To find the slope of a line in the form \( Ax + By + C = 0 \), rearrange it to \( y = mx + b \). - For two lines to be perpendicular, the product of their slopes must equal \(-1\). - Solve for the unknown variable by setting up the equation based on the slopes. - Check each option by substituting back into the slope condition to confirm or deny their validity.