Point A (4, 2) divides segment BC in the ratio 2:5. Coordinates of B and C are (2,6), (9,y) respectively. Find the value of y.

Point A (4, 2) divides segment BC in ratio 2:5, where B = (2, 6) and C = (8, y).

When point A divides BC in ratio 2:5, it means BA : AC = 2 : 5.

When a point A(x, y) divides the line segment joining B(x₁, y₁) and C(x₂, y₂) in ratio m:n:

$x = \dfrac{m \times x_2 + n \times x_1}{m + n}$

$y = \dfrac{m \times y_2 + n \times y_1}{m + n}$

Given values:

A = (4, 2)

B = (2, 6), so $x_1 = 2, y_1 = 6$

C = (8, y), so $x_2 = 8, y_2 = y$

Ratio m:n = 2:5

Using the y-coordinate formula:

$2 = \dfrac{2 \times y + 5 \times 6}{2 + 5}$

$2 = \dfrac{2y + 30}{7}$

$14 = 2y + 30$

$14 - 30 = 2y$

$-16 = 2y$

$y = -8$

Therefore, the value of y is $-8$.

Answer: Option 4