A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?
equition of line passing trough A and B "-7x + 3y = -21.5" can also be written as "y=\\frac{7}{3}x-\\frac{21.5}{3}"
comparing with standard form "y=mx+c"
slope "m=\\frac{7}{3}"
PQ is perpendicular to lane AB. Product of slope of perpendicular lines will be -1.
"m_1.m_2=-1"
slope of the line"=\\frac{-1}{slope \\space of\\space parallel\\space line}=\\frac{-1}{\\frac{7}{3}}=\\frac{-3}{7}"
equition of the line will then be
"y=mx+c\\\\y=-\\frac{3}{7}x+c\\\\7y=-3x+7c\\\\7y+3x=7c"
dividing by 2
"3.5y+1.5x=3.5c"
find c
the line pq passes through the point (7,6) in as shown in the figure hence line could be found out using point slope form of line.
"y-y_1=m(x-4)\\\\slope=\\frac{-3}{7}\\space (7,6)\\\\y-6=\\frac{-3}{7}(x-7)=7\\\\7(y-6)=-3(x-7)\\\\7y-42-3x+21\\\\=7x+3x=21+42\\\\7x+3x=63" is the eqition of central line.
divide by 2
"3.5y+1.5x=31.5"
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