Question

the mid point of (-5, 12) (9,-2) divides the join of the points (-8,-5) ; (7,10) in the ratio ?

Answer 1

Let mid point of A(-5, 12) , B(9,-2) be ~ P(x,y) and it divides the join of the points M(-8,-5) and N(7,10) in the ratio of

$\large\bold{m\::\:n}$

Now,

$\bold{( x , y ) \rightarrow \: ( \frac{ - 5 + 9}{2} , \frac{12 + ( - 2)}{2} )}$

$\: \: \: \: \: \: \: \: \: \: \bold{ \rightarrow \: ( \frac{ - 4}{2} , \frac{12 - 2}{2} )}$

$\: \: \: \: \: \: \: \: \: \: \bold{ \rightarrow \: ( { - 2} , \frac{10}{2} )}$

$\: \: \: \: \: \: \: \: \: \: \bold{ \rightarrow \: ( { - 2} , 5)}$

So, coordinate point of P is (-2 , 5 )

A/Q,

$\bold{(-2 , 5 ) \: \rightarrow( \frac{7m + ( - 8)n}{m + n} , \: \frac{10m + ( - 5)n}{m + n} )}$

$\bold{ \: \: \: \: \: \: \: \: \: \: \: \: \rightarrow( \frac{7m - 8n}{m + n} , \: \frac{10m - 5n}{m + n} )}$

Now,

$\: \: \bold{ - 2 = \frac{7m - 8n}{ m + n} }$

$\: \: \bold{ \implies \: - 2m - 2n = {7m - 8n}{ } }$

$\: \: \bold{ \implies \: 8n - 2n = {7m + 2m}{ } }$

$\: \: \bold{ \implies \: 6n =9m}$

$\: \: \bold{ \implies \: \frac{m}{n} = \frac{6}{9} }$

$\: \: \bold{ \implies \: \frac{m}{n} = \frac{2}{3} }$

$\: \: \bold{ \implies \: \large\boxed{\bold{ m :n = {2} : {3} } }}$

$\Large{\colorbox{yellow}{\underline{\underline{♣Answer♣:—\:\:2\:\:: \: 3} }}}$