To determine which of the given equations represents the central street PC, we need to analyze the relationships between the given equation of the lane passing through [tex]$A$[/tex] and [tex]$B$[/tex] and the possible equations for PC. The central street should be either parallel or perpendicular to the lane passing through [tex]$A$[/tex] and [tex]$B$[/tex].
The equation of the lane passing through [tex]$A$[/tex] and [tex]$B$[/tex] is given as:
[tex]\[
-7x + 3y = -215
\][/tex]
First, we need to determine the slope of this equation. To do that, we rewrite it in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
3y = 7x - 215
\][/tex]
[tex]\[
y = \frac{7}{3}x - \frac{215}{3}
\][/tex]
From this, we see that the slope [tex]\(m_1\)[/tex] of the line passing through [tex]$A$[/tex] and [tex]$B$[/tex] is:
[tex]\[
m_1 = \frac{7}{3}
\][/tex]
Next, we analyze the slopes of the possible equations for the central street PC to see if they are either parallel or perpendicular to the lane passing through [tex]$A$[/tex] and [tex]$B$[/tex].
Option A: [tex]\( -3x + 4y = 3 \)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
4y = 3x + 3
\][/tex]
[tex]\[
y = \frac{3}{4}x + \frac{3}{4}
\][/tex]
The slope [tex]\(m_2\)[/tex] is:
[tex]\[
m_2 = \frac{3}{4}
\][/tex]
Option B: [tex]\( 3x + 7y = 63 \)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
7y = -3x + 63
\][/tex]
[tex]\[
y = -\frac{3}{7}x + 9
\][/tex]
The slope [tex]\(m_2\)[/tex] is:
[tex]\[
m_2 = -\frac{3}{7}
\][/tex]
Option C: [tex]\( 2x + y = 20 \)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
y = -2x + 20
\][/tex]
The slope [tex]\(m_2\)[/tex] is:
[tex]\[
m_2 = -2
\][/tex]
Option D: [tex]\( 7x + 3y = 70 \)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
3y = -7x + 70
\][/tex]
[tex]\[
y = -\frac{7}{3}x + \frac{70}{3}
\][/tex]
The slope [tex]\(m_2\)[/tex] is:
[tex]\[
m_2 = -\frac{7}{3}
\][/tex]
For the central street to be perpendicular to the line passing through [tex]$A$[/tex] and [tex]$B$[/tex], the product of their slopes should be [tex]\(-1\)[/tex]:
[tex]\[
m_1 \cdot m_2 = -1
\][/tex]
Thus,
[tex]\[
\frac{7}{3} \cdot m_2 = -1 \implies m_2 = -\frac{3}{7}
\][/tex]
Comparing this with the slopes of the equations provided, we see that the slope of the line in Option B: [tex]\( 3x + 7y = 63 \)[/tex] is [tex]\( -\frac{3}{7} \)[/tex], which satisfies the condition for perpendicularity.
Therefore, the equation of the central street PC is:
[tex]\[
\boxed{3x + 7y = 63}
\][/tex]
