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What are the necessary criteria for a line to be perpendicular to the given line and have the same [tex]\(y\)[/tex]-intercept?

A. The slope is [tex]\(-\frac{3}{2}\)[/tex] and contains the point [tex]\((0,2)\)[/tex].
B. The slope is [tex]\(-\frac{2}{3}\)[/tex] and contains the point [tex]\((0,-2)\)[/tex].
C. The slope is [tex]\(\frac{3}{2}\)[/tex] and contains the point [tex]\((0,2)\)[/tex].
D. The slope is [tex]\(-\frac{3}{2}\)[/tex] and contains the point [tex]\((0,-2)\)[/tex].

Sagot :

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.