Given that two perpendicular lines intersect at the origin and the slope of the first line is [tex]\(\frac{-1}{2}\)[/tex], we are to find the equation of the second line.
1. Determine the slope of the second line:
- Perpendicular lines have slopes that are negative reciprocals of each other.
- The slope of the first line is [tex]\(\frac{-1}{2}\)[/tex].
- Therefore, to find the slope of the second line, we need to take the negative reciprocal of [tex]\(\frac{-1}{2}\)[/tex]:
[tex]\[
\text{Slope of the second line} = -\left(\frac{1}{\left(\frac{-1}{2}\right)}\right) = -\left(-2\right) = 2
\][/tex]
2. Formulate the equation of the second line:
- The second line passes through the origin, meaning its equation will be in the form [tex]\(y = mx\)[/tex], where [tex]\(m\)[/tex] is the slope.
- Given that the slope of the second line is [tex]\(2\)[/tex], the equation of the second line is:
[tex]\[
y = 2x
\][/tex]
3. Select the correct answer:
- The choices provided are:
A. [tex]\(y=-2x\)[/tex]
B. [tex]\(y=\frac{1}{2}x\)[/tex]
C. [tex]\(y=-\frac{1}{2}x\)[/tex]
D. [tex]\(y=2x\)[/tex]
- From our calculations, the correct equation for the second line is [tex]\(y=2x\)[/tex].
Therefore, the best answer from the provided choices is:
[tex]\[
\boxed{D}
\][/tex]
