Question

select the correct answer. which equation represents the line that is perpendicular to ( y = \frac{1}{4} ) and passes through (-6,-9)? a. ( x = -9 ) b. ( x = -6 ) c. ( y = -9 ) d. ( y = -4 )

Explanation:

Step1: Analyze the given line

The line \( y = \frac{1}{4} \) is a horizontal line (since it has the form \( y = k \), where \( k \) is a constant). A line perpendicular to a horizontal line is a vertical line. Vertical lines have the equation \( x = h \), where \( h \) is the x - coordinate of any point on the line.

Step2: Determine the equation of the perpendicular line

The perpendicular line passes through the point \( (-6, -9) \). For a vertical line passing through \( (-6, -9) \), the x - coordinate is always \( -6 \)? Wait, no, wait. Wait, the original line is \( y=\frac{1}{4}\), which is horizontal. A line perpendicular to a horizontal line is vertical, so its equation is of the form \( x = c \), where \( c \) is the x - coordinate of the point it passes through. Wait, the point is \( (-6, -9) \), so the x - coordinate is \( -6 \)? But wait, looking at the options, option B is \( x=-6 \), but wait, maybe I misread the point. Wait, the problem says "passes through (-6, -9)"? Wait, no, maybe the point is (-6, -9)? Wait, no, let's re - check. Wait, the line \( y=\frac{1}{4}\) is horizontal, slope \( m = 0 \). A line perpendicular to it has an undefined slope (vertical line), so equation \( x = c \), where \( c \) is the x - coordinate of the point. If the point is \( (-6, -9) \), then \( x=-6 \), which is option B. But wait, maybe the point is \( (-9, -4) \)? Wait, no, the problem says "passes through (-6, -9)"? Wait, maybe there's a typo, but looking at the options, option B is \( x = - 6 \), option A is \( x=-9 \), option C is \( y=-9 \), option D is \( y = - 4 \). Wait, no, let's re - derive. The line \( y=\frac{1}{4}\) is horizontal (slope 0). A line perpendicular to a horizontal line is vertical, so its equation is \( x = k \), where \( k \) is the x - coordinate of the point it passes through. If the point is \( (-6, -9) \), then \( x=-6 \) (option B). But wait, maybe the point is \( (-9, -9) \)? No, the problem says "passes through (-6, -9)". Wait, maybe I made a mistake. Wait, no, let's check the options again. Option B is \( x=-6 \), which is a vertical line, perpendicular to \( y=\frac{1}{4}\) (horizontal line), and passes through \( (-6, -9) \) (since for \( x=-6 \), y can be any value, including - 9). So the equation of the perpendicular line is \( x=-6 \), which is option B. Wait, but wait, maybe the point is \( (-9, -9) \)? No, the problem states "passes through (-6, -9)". So the correct equation is \( x=-6 \), which is option B.

Wait, no, wait a second. Wait, maybe the original line is \( y=\frac{1}{4}x \)? No, the problem says \( y = \frac{1}{4} \), which is a horizontal line (constant y - value). So a line perpendicular to it is vertical, so \( x = c \), where \( c \) is the x - coordinate of the point. If the point is \( (-6, -9) \), then \( x=-6 \), so option B.

Answer:

B. \( x = - 6 \)