Question: 23. Consider the line \[ y = -\frac{1}{2}x - 6. \] Find the equation of the line perpendicular…

23. Consider the line

\[ y = -\frac{1}{2}x - 6. \]

Find the equation of the line perpendicular to this line that passes through the point (8, 2).

• \( y = -\frac{1}{2}x - 8 \)
• \( y = \frac{1}{2}x + 2 \)
• \( y = 2x - 14 \)
• \( y = 2x + 14 \)

Solution

To find the equation of the line that is perpendicular to \( y = -\frac{1}{2}x - 6 \) and passes through the point \( (8, 2) \), follow these steps: The slope of the given line is \( -\frac{1}{2} \). The slope of a line perpendicular to another is the negative reciprocal. So, the slope of the perpendicular line is: \[ 2 \] Using the point-slope formula \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point \( (8, 2) \): Substitute \( m = 2 \), \( x_1 = 8 \), and \( y_1 = 2 \): \[ y - 2 = 2(x - 8) \] Simplify the equation: \[ y - 2 = 2x - 16 \] Add 2 to both sides: \[ y = 2x - 14 \] Thus, the equation of the line is \( y = 2x - 14 \). The correct answer is y = 2x - 14.