Question

Find the harmonic conjugate of $(2,1)$ with respect to $(4,2)$ and $(6,3)$

Answer 1

The ratio in which $P(2,1)$ divides the line segment $A(4,2)$ and $B(6,3)$ is $x_1-x:x-x_2$

$=4-2:2-6=2:-4$

The point $Q$ divides $AB$ in the ratio $-m:n=-2:-4=2:4=1:2$

$Q=(\frac{1\times 5+2\times 4}{1+2},\frac{1\times 2+2\times 2}{1+2})=(\frac{13}{3},\frac{6}{3})=(\frac{13}{3},2)$

72)If $A=(-3,4),B=(2,1),$ then find the coordinates of the point $C$ on $AB$ produced such that $AC=2BC$

Given:$AC=2BC$

$\Rightarrow \frac{AC}{BC}=\frac{2}{1}$

$\therefore C$ divides $AB$ in the ratio $2:1$

Using section formula, we have

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Given:$A=(-3,4),B=(2,1)$ and $m=2,n=1$

$\therefore C=(\frac{2\times 2+1\times -3}{2+1},\frac{2\times 1+1\times 4}{2+1})$

$=(\frac{4-3}{3},\frac{2+4}{3})$

$=(\frac{1}{3},\frac{6}{3})$

$\therefore C(x,y)=(\frac{1}{3},2)$