Let's solve the given problem step by step.
### Part (a) Finding the equation of the line [tex]\( l \)[/tex]
First, we need to find the slope of the line segment [tex]\( PQ \)[/tex].
1. Slope of [tex]\( PQ \)[/tex]:
The slope [tex]\( m_{PQ} \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of points [tex]\(P(-4, 2)\)[/tex] and [tex]\(Q(5, -4)\)[/tex]:
[tex]\[
m_{PQ} = \frac{-4 - 2}{5 - (-4)} = \frac{-6}{9} = -\frac{2}{3}
\][/tex]
So, the slope of [tex]\( PQ \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex].
2. Slope of line [tex]\( l \)[/tex] perpendicular to [tex]\( PQ \)[/tex]:
If two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex]. Let the slope of line [tex]\( l \)[/tex] be [tex]\( m_l \)[/tex]. Thus,
[tex]\[
m_l \times m_{PQ} = -1 \implies m_l \times -\frac{2}{3} = -1
\][/tex]
Solving for [tex]\( m_l \)[/tex]:
[tex]\[
m_l = \frac{3}{2}
\][/tex]
So, the slope of line [tex]\( l \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].
3. Finding the y-intercept [tex]\( b \)[/tex]:
The equation of the line [tex]\( l \)[/tex] can be written in slope-intercept form:
[tex]\[
y = m_l x + b
\][/tex]
We use point [tex]\( P (-4, 2) \)[/tex] to find [tex]\( b \)[/tex]:
[tex]\[
2 = \frac{3}{2} \times (-4) + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
2 = -6 + b \implies b = 2 + 6 = 8
\][/tex]
Therefore, the equation of the line [tex]\( l \)[/tex] is:
[tex]\[
y = \frac{3}{2}x + 8
\][/tex]
### Part (b) Finding the coordinates of point [tex]\( R \)[/tex]
Point [tex]\( R \)[/tex] is the y-intercept of the line [tex]\( l \)[/tex], which means it lies on the y-axis. For any point on the y-axis, the [tex]\( x \)[/tex]-coordinate is 0.
1. Substituting [tex]\( x = 0 \)[/tex] in the equation of line [tex]\( l \)[/tex]:
[tex]\[
y = \frac{3}{2}(0) + 8 = 8
\][/tex]
Thus, the coordinates of point [tex]\( R \)[/tex] are:
[tex]\[
R = (0, 8)
\][/tex]
### Summary
1. The equation of the line [tex]\( l \)[/tex] is:
[tex]\[
y = \frac{3}{2}x + 8
\][/tex]
2. The coordinates of the point [tex]\( R \)[/tex] are:
[tex]\[
R = (0, 8)
\][/tex]
