Question

1. In what ratio is the line joining the points $A(4,4)$ and $B(7,7)$ divided by point $P(-1,-1)$.
2. Find the ratio in which the point $P(m,6)$ divides the join of $A(-4,3)$ and $B(2,8)$.Also, find m.

Answer 1

$\left(1\right)$

$AP:PB=m:n$

external division

$(4,4)(7,7)$

$x=\frac{7m{k}_2-n{k}_1}{m-n}$

$-1=\frac{7m-4n}{m-n}$

$7m-4n=-m+n$

$8m=5n$

$\frac{m}{n}=\frac{5}{8}$

$\left(2\right)$

$P(m,6)$

$A(-4,3)B(2,8)$

Let $k:1$ be ratio

$y=\frac{k\left(8\right)+1\left(3\right)}{k+1}$

$[\therefore y=\frac{m{y}_2+n{y}_1}{m+n}]$

$(k+1{)}^6=8k+3$

$8k+3=6k+6\to k=\frac{3}{2}$

Ratio is $3:2$