If the points are collinear and 2b + c = 4, find the values of b and c.

Given points A(-1, -4), B(b, c) and C(5, -1) are collinear

And 2b + c = 4 … (1)

We have to find the values of a and b.

Since the given points are collinear, the area of the triangle formed by them must be 0.

We know that the area of Triangle =

∴ = 0

⇒ [x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)] = 0

⇒ [-1(c – (-1)) + b(-1 – (-4)) + 5(-4 – c)] = 0

⇒ [-1(c + 1) + 3b + 5(-4 – c)] = 0

⇒ [-c - 1 + 3b - 20 – 5c] = 0

⇒ 3b – 6c - 21 = 0

⇒ 3b – 6c = 21

Dividing by 3, we get

⇒ b – 2c = 7

⇒ b = 7 + 2c … (2)

Substituting (2) in (1),

⇒ 2(7 + 2c) + c = 4

⇒ 14 + 4c + c = 4

⇒ 5c = -10

∴ c = -2

Substituting value of c in (1),

⇒ 2b + (-2) = 4

⇒ 2b = 6

∴ b = 3

The values of a and b are 3 and -2 respectively.