Question

Which of the following are perpendicular to y= 1/2x and pass through the point (1,3)? select all the apply

A. y=-2x+5
B. y= 1/2x+5/2
C. y+1=-2(x-3)
D. y-2=1/2(x+1)
E. 2x-y=5

Answer 1

To find the lines that are perpendicular to \( y = \frac{1}{2}x \) and pass through the point (1, 3), we first need to determine the slope of the line \( y = \frac{1}{2}x \), which is \(\frac{1}{2}\).

Lines that are perpendicular to a line have slopes that are negative reciprocals of the line's slope. Therefore, the slope of a line perpendicular to \( y = \frac{1}{2}x \) is:

\[
-\frac{1}{\left(\frac{1}{2}\right)} = -2
\]

Next, we will determine which of the given options has a slope of \(-2\) and passes through the point (1, 3).

**Option A: \( y = -2x + 5 \)**
- The slope is \(-2\).
- To check if it passes through (1, 3):
\[ y = -2(1) + 5 = -2 + 5 = 3 \]
Therefore, yes, it passes through (1, 3).

**Option B: \( y = \frac{1}{2}x + \frac{5}{2} \)**
- The slope is \(\frac{1}{2}\), which is not perpendicular.
- Therefore, it does not qualify.

**Option C: \( y + 1 = -2(x - 3) \)**
- Rewrite in slope-intercept form:
\[ y + 1 = -2x + 6 \]
\[ y = -2x + 5 \]
- The slope is \(-2\).
- To check if it passes through (1, 3):
\[ y = -2(1) + 5 = -2 + 5 = 3 \]
Therefore, yes, it passes through (1, 3).

**Option D: \( y - 2 = \frac{1}{2}(x + 1) \)**
- Rewrite in slope-intercept form:
\[ y - 2 = \frac{1}{2}x + \frac{1}{2} \]
\[ y = \frac{1}{2}x + \frac{5}{2} \]
- The slope is \(\frac{1}{2}\), which is not perpendicular.
- Therefore, it does not qualify.

**Option E: \( 2x - y = 5 \)**
- Rearranging gives \( y = 2x - 5 \).
- The slope is \(2\), which is not perpendicular.
- Therefore, it does not qualify.

Based on the analysis, the options that are perpendicular to \( y = \frac{1}{2}x \) and pass through the point (1, 3) are:

- **A. \( y = -2x + 5 \)**
- **C. \( y + 1 = -2(x - 3) \)**