Question

find the ratio in which line 3x+2y=17 divide the line segment joint by point (2, 5)and (5,2)​

Answer 1

$\textbf{Given:}$

$\textsf{Points are (2,5) and (5,2)}$

$\textbf{To find:}$

$\textsf{The ratio in which the line 3x+2y-17=0 divides the}$

$\textsf{segment joining (2,5) and (5,2)}$

$\textbf{Solution:}$

$\textsf{The coordinats of the point which divides the line segment joining}$

$\textsf{(2,5) and (5,2) internally in the ratio m:n are}$

$\mathsf{\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)}$

$\mathsf{\left(\dfrac{m(5)+n(2)}{m+n},\dfrac{m(2)+n(5)}{m+n}\right)}$

$\mathsf{\left(\dfrac{5m+2n}{m+n},\dfrac{2m+5n}{m+n}\right)}$

$\textsf{But, this point lies on the line 3x+2y-17=0}$

$\mathsf{3\left(\dfrac{5m+2n}{m+n}\right)+2\left(\dfrac{2m+5n}{m+n}\right)-17=0}$

$\mathsf{\dfrac{15m+6n}{m+n}+\left(\dfrac{4m+10n}{m+n}\right)-17(m+n)=0}$

$\mathsf{(15m+6n)+(4m+10n)-17(m+n)=0}$

$\mathsf{19m+16n-17m-17n=0}$

$\mathsf{2m-n=0}$

$\mathsf{2m=n}$

$\mathsf{\dfrac{m}{n}=\dfrac{1}{2}}$

$\therefore\underline{\textsf{The line divides the line joining the given points}}$

$\underline{\textsf{internally in the ratio 1:2}}$