In which ratio does the point $P (2, b)$ divide the line segment joining $A (1, 2)$ and $B (4, 5)$ from $A$? Also,find the value of $b$.

(N/A) Let the point $P (2, b)$ divide the line segment joining $A (1, 2)$ and $B (4, 5)$ in the ratio $k: 1$.
Using the section formula for the $x$-coordinate: $x = \frac{mx_2 + nx_1}{m + n}$.
Substituting the values: $2 = \frac{k(4) + 1(1)}{k + 1}$.
$2(k + 1) = 4k + 1 \implies 2k + 2 = 4k + 1 \implies 2k = 1 \implies k = \frac{1}{2}$.
Thus,the ratio is $1: 2$.
Now,find the $y$-coordinate using the section formula: $y = \frac{my_2 + ny_1}{m + n}$.
$b = \frac{1(5) + 2(2)}{1 + 2} = \frac{5 + 4}{3} = \frac{9}{3} = 3$.
Therefore,the ratio is $1: 2$ and $b = 3$.