Certainly! Let's solve this step-by-step.
1. Identify the slope of the given path:
The given path is represented by the equation \( y = -4x - 6 \). The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope.
Therefore, the slope (\( m_1 \)) of the given path is \( -4 \).
2. Find the slope of the new path:
Since the new path is perpendicular to the given path, its slope (\( m_2 \)) will be the negative reciprocal of the slope of the given path.
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-4} = \frac{1}{4}
\][/tex]
3. Identify the point of intersection:
The paths intersect at the point \((-4, 10)\).
4. Use the point-slope form of the equation:
The point-slope form of a line passing through a point \((x_1, y_1)\) with slope \( m \) is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plugging in the slope (\( m_2 = \frac{1}{4} \)) and the point \((-4, 10)\):
[tex]\[
y - 10 = \frac{1}{4}(x + 4)
\][/tex]
5. Simplify the equation:
First, distribute the \(\frac{1}{4}\) on the right-hand side:
[tex]\[
y - 10 = \frac{1}{4}x + 1
\][/tex]
Then, add 10 to both sides to solve for \( y \):
[tex]\[
y - 10 + 10 = \frac{1}{4}x + 1 + 10
\][/tex]
[tex]\[
y = \frac{1}{4}x + 11
\][/tex]
So, the equation that represents the new path is:
[tex]\[
\boxed{y = \frac{1}{4}x + 11}
\][/tex]
Therefore, the correct answer is D. [tex]\( y = \frac{1}{4}x + 11 \)[/tex].
