Question

In what ratio is the line joining P(-3,5) and Q (4, -9) divided by the point M (x,-5) ? Also find the value of x​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{P(-3,5) and Q(4,-9)}$

$\underline{\textbf{To find:}}$

$\textsf{(i)The ratio in which the line joining P and Q}$$\;\textsf{is divided by the point (x,-5)}$

$\textsf{(ii) The value of x}$

$\underline{\textbf{Solution:}}$

$\textsf{Let m:n be the ratio of the line joining P and Q is divided by (x,-5)}$

$\textsf{By Section formula, we get}$

$\mathsf{\left(\dfrac{m\,x_2+n\,x_1}{m+n},\dfrac{m\,y_2+n\,y_1}{m+n}\right)=(x,-5)}$

$\mathsf{\left(\dfrac{m(4)+n(-3)}{m+n},\dfrac{m(-9)+n(5)}{m+n}\right)=(x,-5)}$

$\mathsf{\left(\dfrac{4m-3n}{m+n},\dfrac{-9m+5n}{m+n}\right)=(x,-5)}$

$\textsf{Equatting the corresponding co-ordinates}$

$\textsf{on bothsides, we get}$

$\mathsf{\dfrac{4m-3n}{m+n}=x\;\;\;\&\;\;\;\dfrac{-9m+5n}{m+n}=-5}$

$\mathsf{\dfrac{-9m+5n}{m+n}=-5}$

$\implies\mathsf{-9m+5n=-5(m+n)}$

$\implies\mathsf{-9m+5n=-5m-5n}$

$\implies\mathsf{-9m+5m=-5n-5n}$

$\implies\mathsf{-4\,m=-10\,n}$

$\implies\mathsf{\dfrac{m}{n}=\dfrac{-10}{-4}}$

$\implies\mathsf{\dfrac{m}{n}=\dfrac{5}{2}}$

$\implies\boxed{\textbf{m:n=5:2}}$

$\mathsf{Also,}$

$\mathsf{\dfrac{4(5)-3(2)}{5+2}=x}$

$\mathsf{\dfrac{20-6}{7}=x}$

$\mathsf{\dfrac{14}{7}=x}$

$\implies\boxed{\mathsf{x=2}}$