In what ratio does the point P(2, 5) divide the join of A(8, 2) and B(−6, 9)?

Let the point P(2, 5) divide AB in the ratio k : 1.
Then, by section formula, the coordinates of P are

$x=\frac{-6 k+8}{k+1}, y=\frac{9 k+2}{k+1}$

It is given that the coordinates of $P$ are $P(2,5)$.

$\Rightarrow 2=\frac{-6 k+8}{k+1}, 5=\frac{9 k+2}{k+1}$

$\Rightarrow 2 k+2=-6 k+8,5 k+5=9 k+2$

$\Rightarrow 2 k+6 k=8-2,5-2=9 k-5 k$

$\Rightarrow 8 k=6,4 k=3$

$\Rightarrow k=\frac{6}{8}, k=\frac{3}{4}$

$\Rightarrow k=\frac{3}{4}$ in each case.

Therefore, the point P(2, 5) divides AB in the ratio 3 : 4.