To solve this problem, we need to determine the criteria for a line to be perpendicular to a given line and to have the same [tex]\( y \)[/tex]-intercept.
1. Understanding perpendicular slopes:
- Given the slope of the original line is [tex]\(-\frac{3}{2}\)[/tex], the slope of a line perpendicular to it will be the negative reciprocal of the original slope.
- The negative reciprocal of [tex]\( -\frac{3}{2} \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
2. Identifying the [tex]\( y \)[/tex]-intercept:
- The [tex]\( y \)[/tex]-intercept of the original line is given as 2.
- Therefore, the line we are looking for must also cross the [tex]\( y \)[/tex]-axis at [tex]\( y = 2 \)[/tex].
3. Checking the criteria for each option:
- Option 1: The slope is [tex]\( -\frac{3}{2} \)[/tex] and contains the point [tex]\( (0,2) \)[/tex].
- This line is not perpendicular to the given line because it has the same slope as the original line.
- Option 2: The slope is [tex]\( -\frac{2}{3} \)[/tex] and contains the point [tex]\( (0,-2) \)[/tex].
- This line has a slope that is not the negative reciprocal of the original line’s slope, and it does not have the same [tex]\( y \)[/tex]-intercept as the original line.
- Option 3: The slope is [tex]\( \frac{3}{2} \)[/tex] and contains the point [tex]\( (0,2) \)[/tex].
- This line has a slope that is not the negative reciprocal of the original line’s slope, even though it has the correct [tex]\( y \)[/tex]-intercept.
- Option 4: The slope is [tex]\( -\frac{3}{2} \)[/tex] and contains the point [tex]\( (0,-2) \)[/tex].
- This line has the same slope as the original line, and it does not have the same [tex]\( y \)[/tex]-intercept.
4. The correct criteria:
- The slope should be [tex]\( \frac{2}{3} \)[/tex] (the negative reciprocal of [tex]\(-\frac{3}{2}\)[/tex]).
- The [tex]\( y \)[/tex]-intercept should be 2.
Therefore, none of the provided options meet the required conditions for the line to be perpendicular to the given line and have the same [tex]\( y \)[/tex]-intercept. The appropriate criteria for the line are:
- The slope is [tex]\( \frac{2}{3} \)[/tex] and it contains the point [tex]\( (0, 2) \)[/tex].
