Question

If the point (x,y),(1,-2),(3,-4) are collinear prove that x+y+1=0

Answer 1

Let

(x, y) = (a, b)

(1, -2) = (c, d)

(3, -4) = (e, f)

since the points are collinear

therefore, area of the triangle formed by these points is equal to zero

ar(∆) = 0

$0 = \frac{1}{2} |a(d - f) + c(f - b) + e(b - d)| \\ 0 \times 2 = |x( - 2 + 4) + 1( - 4 - y) + 3(y + 2)| \\ 0 = 2x - 4 - y + 3y + 6 \\ 0 = 2x + 2y + 2 \\ \\ dividing \: throughout \: by \: 2 \\ \\ 0 = x + y + 1$

Hence, x + y + 1 = 0.