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Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.

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#### Solution

Given, the points are (a + 5, a – 4), (a – 2, a + 3) and (a, a).

We have to prove that these pints do not lie on a straaighine.

So, we have to prove that these points form a triangle.

Area, Δ = `1/2|("a" + 5, "a" - 4, 1),("a" - 2, "a" + 3, 1),("a", "a", 1)|`

[Applying R_{1} → R_{1} – R_{3} and R_{2} → R_{2} – R_{3}]

= `1/2 |(5, -4, 0),(-2, 3, 0),("a", "a", 1)|`

= `1/2[(1 * (15 - 8)]`

= `7/2 ≠ 0`

Hence, given points from a triangle i.e., points do not lie on a straight line.

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