Let A(x_{1}, y_{1}) and B(x_{2}, y_{2}) be two given distinct points on a line as shown below.
Slope of the straight line passing through these points is given by
Point-Slope form equation of a line,
y - y_{1} = m(x - x_{1})
The above is the equation of a line in two-point form.
Example 1 :
Find the general equation of a line passing through the points (-2, 1) and (4, -7).
Solution :
Given : Two points on the straight line : (-2, 1) and (4, -7).
Equation of line in two-point form :
y - y_{1} = [(y_{2} - y_{1})/(x_{2} - x_{1})](x - x_{1})
Substitute (x_{1} , y_{1}) = (-2, 1) and (x_{2}, y_{2}) = (4, -7).
y - 1 = [(-7 - 1)/(4 + 2)](x + 2)
y - 1 = (-8/6)(x + 2)
y - 1 = -(4/3)(x + 2)
3(y - 1) = -4(x + 2)
3y - 3 = -4x - 8
4x + 3y + 5 = 0
Example 2 :
Find the equation of a line in slope-intercept form which passing through the points (-2, 5) and (3, 6).
Solution :
Given : Two points on the straight line : (-2, 5) and (3, 6).
Equation of line in two-point form :
y - y_{1} = [(y_{2} - y_{1})/(x_{2} - x_{1})](x - x_{1})
Substitute (x_{1} , y_{1}) = (-2, 5) and (x_{2}, y_{2}) = (3, 6).
y - 5 = [(6 - 5)/(3 + 2)](x + 2)
y - 5 = (1/5)(x + 2)
y - 5 = (1/5)x + 2/5
y = (1/5)x + 2/5 + 5
y = (1/5)x + 27/5
Example 3 :
The vertices of a triangle ABC are A(2, 1), B(-2, 3) and C(4, 5). Find the equation of the median through the vertex A.
Solution :
Median is a straight line joining a vertex and the midpoint of the opposite side.
In ΔABC above, midpoint of BC :
= D((-2 + 4)/2, (3 + 5)/2)
= D(1, 4)
The median through A is the line joining two points A (2, 1) and D(1, 4).
Equation of the median through A :
y - y_{1} = [(y_{2} - y_{1})/(x_{2} - x_{1})](x - x_{1})
Substitute (x_{1} , y_{1}) = (2, 1) and (x_{2}, y_{2}) = (1, 4).
y - 1 = [(4 - 1)/(1 - 2)](x - 2)
y - 1 = (3/-1)(x - 2)
y - 1 = -3(x - 2)
y - 1 = -3x + 6
3x + y - 7 = 0
Example 4 :
Two buildings of different heights are located at opposite sides of each other. If a heavy rod is attached joining the terrace of the buildings from (6, 10) to (14, 12), find the equation of the rod joining the buildings ?
Solution :
The equation of the rod is the equation of the line passing through the two points (6, 10) and (14, 12).
Equation of the line in two-point form :
y - y_{1} = [(y_{2} - y_{1})/(x_{2} - x_{1})](x - x_{1})
Substitute (x_{1} , y_{1}) = (6, 10) and (x_{2}, y_{2}) = (14, 12).
y - 10 = [(12 - 10)/(14 - 6)](x -6)
y - 10 = (2/8)(x - 6)
y - 10 = (1/4)(x - 6)
4(y - 10) = x - 6
4y - 40 = x - 6
x - 4y + 34 = 0
Hence, equation of the rod is x - 4y + 34 = 0.
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