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The problem asks to construct the equation of a line that goes through the origin and is perpendicular to two given equations:

- \( a. \ y = 8 \) (a horizontal line)
- \( b. \ x = -4 \) (a vertical line)

Let's break this down for both parts:

### Part a: \( y = 8 \)
The equation \( y = 8 \) represents a horizontal line, which has a slope of 0 (because it is constant along the y-axis). The line perpendicular to a horizontal line must be vertical. A vertical line passing through the origin has the equation:

\[ x = 0 \]

Thus, the equation of the line perpendicular to \( y = 8 \) and passing through the origin is:
\[ x = 0 \]

### Part b: \( x = -4 \)
The equation \( x = -4 \) represents a vertical line, which has an undefined slope. A line perpendicular to a vertical line must be horizontal. The equation of a horizontal line passing through the origin is:

\[ y = 0 \]

Thus, the equation of the line perpendicular to \( x = -4 \) and passing through the origin is:
\[ y = 0 \]

Would you like more details on perpendicular lines, or do you have any questions? Here are 5 related questions you might find helpful:

1. How do you determine the slope of a line from its equation?
2. What is the general form of the equation of a line?
3. How do perpendicular slopes relate to each other mathematically?
4. What is the significance of the y-intercept in linear equations?
5. Can you explain why vertical lines have an undefined slope?

**Tip:** Remember, two lines are perpendicular if the product of their slopes equals \(-1\), except when one of the lines is vertical (undefined slope) and the other is horizontal (zero slope).

Construct the equation of a line that goes through the origin and is perpendicular to the given equation:

a. y = 8
b. x = -4

Find the equations of lines that pass through the origin and are perpendicular to y = 8 (a horizontal line) and x = -4 (a vertical line). This problem covers key concepts in coordinate geometry, including slopes and perpendicular lines. Suitable for students in Grades 9-10.

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