Question

If the points A (-1, -4), B (b,c) and C (5, -1) are collinear and 2b + c =4, find the values of b and c.

Answer 1

If the area of a triangle formed by three points is zero, Hence the points are collinear.

Area of a triangle whose sides are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$
Area = $\frac{1}{2}[(x_1{y}_2-x_2{y}_1)+(x_2{y}_3-x_3{y}_2)+(x_3{y}_1-x_1{y}_3)]$

0 = $\frac{1}{2}$[-c+4b-b-5c-20-1]
-6c-3b-21 = 0
6c+3b +21 = 0
2c+b+7=0 ----- (i)
2b+c -4 =0 ---- (ii)
from (i) and(ii)
-3b+15=0

b =5
c = -6