To determine the equation of the line that is perpendicular to the given line and has the same [tex]\( y \)[/tex]-intercept, we need to follow these steps:
1. Identify the slope of the given line:
The equation of the given line is:
[tex]\[
y = \frac{1}{5}x + 1
\][/tex]
Here, the slope ([tex]\(m\)[/tex]) is [tex]\(\frac{1}{5}\)[/tex].
2. Find the perpendicular slope:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the given line. Therefore, the perpendicular slope is:
[tex]\[
-\frac{1}{\frac{1}{5}} = -5
\][/tex]
3. Identify the [tex]\( y \)[/tex]-intercept of the given line:
The [tex]\( y \)[/tex]-intercept of the given line is 1, as seen in the equation [tex]\(y = \frac{1}{5}x + 1\)[/tex].
4. Formulate the equation of the perpendicular line:
Utilizing the perpendicular slope found in step 2 and the [tex]\( y \)[/tex]-intercept from step 3, the equation of the line perpendicular to the given line, which also has the same [tex]\( y \)[/tex]-intercept, is:
[tex]\[
y = -5x + 1
\][/tex]
5. Compare with the given options:
We need to compare this equation with the options provided to see which matches:
- [tex]\( y = \frac{1}{5}x + 1 \)[/tex] (This has the same [tex]\( y \)[/tex]-intercept but is not perpendicular)
- [tex]\( y = \frac{1}{5}x + 5 \)[/tex] (This has a different [tex]\( y \)[/tex]-intercept and is not perpendicular)
- [tex]\( y = 5x + 1 \)[/tex] (This is not correct; although it has the same [tex]\( y \)[/tex]-intercept, the slope is incorrect)
- [tex]\( y = 5x + 5 \)[/tex] (This also has a different [tex]\( y \)[/tex]-intercept and is not perpendicular)
After considering all the options, none of these match our equation [tex]\(y = -5x + 1\)[/tex].
Hence, there is no equation in the provided options that satisfies the conditions of being perpendicular to the given line [tex]\( y = \frac{1}{5}x + 1 \)[/tex] and having the same [tex]\( y \)[/tex]-intercept. Thus, the correct answer is:
[tex]\[
\boxed{\text{None}}
\][/tex]
