In what ratio does the x-axis divide the join of A(2, −3) and B(5, 6)?

(c) 1 : 2

Let $A B$ be divided by the $x$ axis in the ratio $k: 1$ at the point $P$.

Then, by section formula, the coordinates of P are

$P\left(\frac{5 k+2}{k+1}, \frac{6 k-3}{k+1}\right)$

But P lies on the x axis: so, its ordinate is 0.

$\frac{6 k-3}{k+1}=0$

$\Rightarrow 6 k-3=0$

$\Rightarrow 6 k=3$

$\Rightarrow k=\frac{1}{2}$

Hence, the required ratio is $\frac{1}{2}: 1$, which is same as $1: 2$.