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Question
Find equation of line joining (1, 2) and (3, 6) using the determinant.
Solution
Let there be a point (x, y).
Therefore the vertices of the triangle will be (x, y), (1, 2), (3,6).
`Delta` area of `Delta` = `1/2 abs ((x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1))`
`x_1 = x, y_1 = y, x_2 = 1, y_2 = 2, x_3 = 3, y_3 = 6`
`= 1/2 abs ((x,y,1),(1,2,1),(3,6,1))`
`= 1/2 [x (2 - 6) - y (1 - 3) + 1(6 - 6)]`
`= 1/2 [x xx (-4) - y (-2) + 1 xx 0]`
`= 1/2 [- 4x + 2y]`
`= 1/2 xx 2 (-2x + y)`
`= -2x + y`
The points are collinear.
So, the area of the triangle is
Therefore the area of `Delta` will be zero.
`=> 0 = -2x + y`
`=> 2x - y = 0`
= y = 2x
This is the required equation.
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