To determine the necessary criteria for a line to be perpendicular to a given line and have the same [tex]\( y \)[/tex]-intercept, let's dissect the problem step-by-step.
1. Identify the Slope of the Given Line:
- The original line we are considering has a slope of [tex]\(-\frac{3}{2}\)[/tex] (slope [tex]\( m = -\frac{3}{2}\)[/tex]).
2. Find the Slope of the Perpendicular Line:
- The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
- Therefore, the slope [tex]\(m_p\)[/tex] of the perpendicular line is:
[tex]\[
m_p = -\frac{1}{m} = -\frac{1}{-\frac{3}{2}} = \frac{2}{3}
\][/tex]
3. Check the [tex]\( y \)[/tex]-intercept:
- The given line passes through the point [tex]\((0, 2)\)[/tex], so its [tex]\( y \)[/tex]-intercept is 2.
4. Verify Other Lines:
- Now, we need to check each of the given lines to see if their slope is [tex]\(\frac{2}{3}\)[/tex] and if they pass through the [tex]\( y \)[/tex]-intercept 2.
- Line 1: Slope [tex]\( -\frac{3}{2} \)[/tex] and point [tex]\( (0, 2) \)[/tex]
- Slope does not match the required [tex]\(\frac{2}{3}\)[/tex].
- Line 2: Slope [tex]\( -\frac{2}{3} \)[/tex] and point [tex]\( (0, -2) \)[/tex]
- Slope does not match the required [tex]\(\frac{2}{3}\)[/tex] and the [tex]\( y \)[/tex]-intercept is incorrect.
- Line 3: Slope [tex]\( \frac{3}{2} \)[/tex] and point [tex]\( (0, 2) \)[/tex]
- Slope does not match the required [tex]\(\frac{2}{3}\)[/tex].
- Line 4: Slope [tex]\( -\frac{3}{2} \)[/tex] and point [tex]\( (0, -2) \)[/tex]
- Slope does not match the required [tex]\(\frac{2}{3}\)[/tex] and the [tex]\( y \)[/tex]-intercept is incorrect.
Given this analysis, none of the provided lines meet the necessary criteria of having a perpendicular slope of [tex]\(\frac{2}{3}\)[/tex] and the same [tex]\( y \)[/tex]-intercept of 2.
Therefore, the answer is no line from the given options meets the criteria.
