Question

Determine the ratio in which the point P(3,5) divides the join of A(1,3) & B(7,9). Find the harmonic conjugate of P w.r.t. A & B.

Answer 1

Ratio: 1:2 internally.

Harmonic Conjugate: (-5,-3).

Answer 2

1. Ratio of Division: Use the section formula $P(x,y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)$. Set P as $(x,y)$, A as $(x_1, y_1)$, B as $(x_2, y_2)$, and solve for $m/n$ (or $k$). If $k>0$, it's internal; if $k<0$, it's external.

2. Harmonic Conjugate: If a point P divides a line segment AB in the ratio $k$ (internal or external), its harmonic conjugate Q divides AB in the ratio $-k$ (external or internal, respectively). Apply the section formula again with the new ratio to find Q.