In what ratio does the point (-4, b) divide the line segment joining the points P(2, -2), Q(-14, 6)? Hence, find the value of b.

Let the point P(-4, b) divide the line segment joining the points P(2, -2) and Q(-14, 6) in the ratio m₁ : m₂.

By section formula, the x-coordinate = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}\Big)

Putting value in above formula we get,

$\Rightarrow -4 = \dfrac{m$1 \times -14 + m2 \times 2}{m1 + m2} \\[1em] \Rightarrow -4m1 - 4m2 = -14m1 + 2m2 \\[1em] \Rightarrow -4m1 + 14m1 = 2m2 + 4m2 \\[1em] \Rightarrow 10m1 = 6m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{6}{10} \\[1em] \Rightarrow m1 : m2 = 3 : 5.

By section formula, y-coordinate = $\Big(\dfrac{m$1y2 + m2y1}{m1 + m2}\Big)

Putting value in above formula we get,

$\Rightarrow b = \dfrac{m$1 \times 6 + m2 \times -2}{m1 + m2} \\[1em] = \dfrac{6m1 - 2m2}{m1 + m2} \\[1em] = \dfrac{6 \times 3 - 2 \times 5}{3 + 5} \\[1em] = \dfrac{18 - 10}{8} \\[1em] = \dfrac{8}{8} \\[1em] = 1.

Hence, the ratio is 3 : 5 and the value of b = 1.