Answer: Option B. -1.5x-3.5y=-31.5
Solution:
Equation of the lane passing through A and B: -7x+3y=-21.5
If we isolate y, and express the equation in the slope-intercept form y=mx+b, the coefficient of the variable "x" is the slope of this line:
Adding 7x both sides of the equation:
-7x+3y+7x=-21.5+7x
3y=7x-21.5
Dividing both sides of the equation by 3:
(3/3)y=(7/3)x-21.5/3
y=(7/3)x-21.5/3
The slope of this line is m1=7/3 (coefficient of variable x)
This line is perpendicular to the central line PQ, then the product of their slopes must be equal to -1.
Slope of AB: m1=7/3
Slope of PQ: m2
m1 m2 = -1
Replacing m1=7/3
(7/3) m2 = -1
Solving for m2: Multiplying both sides of the equation by 3/4:
(3/7)(7/3) m2 = (3/7)(-1)
m2 = - 3/7
According with the graph we know that the line PQ passes through the point P=(7,6)=(xp, yp)→xp=7, yp=6
Then, using the point-slope equation:
y-y1=m(x-x1); y1=yp=6, m=m2=-3/7, x1=xp=7
Replacing the values:
y-6=(-3/7)(x-7)
Multiplying both sides of the equation by 7:
7(y-6)=7(-3/7)(x-7)
7y-42=(-3)(x-7)
7y-42=-3x+21
Subtracting 7y and 21 both sides of the equation:
7y-42-7y-21=-3x+21-7y-21
-63=-3x-7y
-3x-7y=-63
Dividing all the terms by 2:
-(3/2)x-(7/2)y=-63/2
-1.5x-3.5y=-31.5
