Question

A line passes through \( (-1, -3) \) and is perpendicular to \( x + 6y = 5 \). Its x-intercept is:

• A. \( -\frac{1}{2} \)
• B. \( -2 \)
• C. \( 2 \)
• D. \( \frac{1}{2} \)

Hint:

To find the x-intercept of a line, set \( y = 0 \) in the equation of the line and solve for \( x \).

Answer 1

The Correct Option is A

The slope of the given line \( x + 6y = 5 \) is obtained by rewriting the equation in slope-intercept form: \[ 6y = -x + 5 \quad \Rightarrow \quad y = -\frac{1}{6}x + \frac{5}{6} \] So, the slope \( m_1 = -\frac{1}{6} \). The slope of the line perpendicular to this line is the negative reciprocal: \[ m_2 = 6 \] Now, the equation of the line passing through \( (-1, -3) \) with slope \( 6 \) is: \[ y + 3 = 6(x + 1) \] Simplifying: \[ y + 3 = 6x + 6 \quad \Rightarrow \quad y = 6x + 3 \] To find the x-intercept, set \( y = 0 \): \[ 0 = 6x + 3 \quad \Rightarrow \quad x = -\frac{1}{2} \] Thus, the x-intercept is \( -\frac{1}{2} \).