Determining the Equation of a Line Given Two Distinct Points in a Plane, High school final essays of Mathematics

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Lines in 𝑅

Any two distinct points 𝑃 1 (𝑥 1 , 𝑦 1 ) and 𝑃 2 (𝑥 2 , 𝑦 2 ) in 𝑅 2 determine a straight line whose equation is ax + by + c = 0 ( 1 )

where a, b, and c are

real numbers, and a

and b are not both

zero.

We now write (1), (2), and (3) as a linear system in the unknown a, b, and c, obtaining x a + y b + c = 0 𝑥 1 𝑎 + 𝑦 1 𝑏 + 𝑐 = 0 𝑥 2 𝑎 + 𝑦 2 𝑏 + 𝑐 = 0 (4)

We seek a condition on the values x and y that allow ( 4 ) to have a nontrivial solution a, b, and c. Since ( 4 ) is a homogeneous system, it has a nontrivial solution if and only if the determinant of the coefficient matrix is zero, that is, if and only if 𝑥 𝑦 1 𝑥

𝑦

1 𝑥

𝑦

1 = 0 (5) Thus every point P(x,y) on the line satisfies (5) and, conversely, a point satisfying (5) lies on the line.

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