Question

find the coordinates of point which divides the line segment joining the points (a+b,a-b) and (a-b,a+b) in the ratio 3:2 interally
​

Answer 1

Solution:

We know, the coordinate of the point P(x, y) which divides the line segment joining A($\sf{x_1, y_1}$) and B($\sf{x_2, y_2}$) in the ratio m : n are given by:

$\sf{x\:=\:\dfrac{mx_2\:+\:nx_1}{m\:+\:n}\:,\:y\:=\:\dfrac{my_2\:+\:ny_1}{m\:+\:n}}$

So here, m : n = 3 : 2

$\:$ (a+b, a-b) = P($\sf{x_1,\:y_1}$)

$\:$ (a-b, a+b) = A($\sf{x_2,\:y_2}$)

$\:$ $\:$ $\:$ $\:$ $:\implies\:\sf{x\:=\:\dfrac{3(a-b)\:+\:2(a+b)}{3\:+\:2}\:,\:y\:=\:\dfrac{3(a+b)\:+\:2(a-b)}{3\:+\:2}}$

$\:$

$\:$ $\:$ $\:$ $\:$ $:\implies\:\sf{x\:=\:\dfrac{3a\:-\:3b\:+\:2a\:+\:2b}{5}\:,\:y\:=\:\dfrac{3a\:+\:3b\:+\:2a\:-\:2b}{5}}$

$\:$

$\:$ $\:$ $\:$ $\:$ $:\implies\:\sf{x\:=\:\dfrac{3a\:+\:2a\:-\:3b\:+\:2b}{5}\:,\:y\:=\:\dfrac{3a\:+\:2a\:+\:3b\:-\:2b}{5}}$

$\:$

$\:$ $\:$ $\:$ $\:$ $:\implies\:\sf{x\:=\:\dfrac{5a\:-\:b}{5}\:,\:y\:=\:\dfrac{5a\:+\:b}{5}}$

$\:$

$\:$ $\:$ $\:$ $\:$ ✯${\small{\boxed{\sf{\implies\:{\bigg(\dfrac{5a\:-\:b}{5},\:\dfrac{5a\:+\:b}{5})}}}}}$

Answer 2

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Coordinate \: of \: O \: = \bigg( \frac{5a - b}{5} . \frac{5a + b}{5} \bigg)}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

• Let AOB be a line segment whose points are A , O and B so that point O divides the line segment into m : n = 3 : 2.

$\green{\underline{\bold{Given :}}} \\ \tt: \implies Coordinate \: of \: A = (a + b,a - b) \\ \\ \tt: \implies Coordinate \: of \: B = (a - b,a + b) \\ \\ \tt: \implies Ratio = 3 : 2 \\ \\ \red{\underline{\bold{To \: Find :}}} \\ \tt: \implies Coordinate \: of \: O =?$

• According to given question :

$\green{ \star} \tt \: \frac{m}{n} = \frac{3}{2} \\ \\ \green{ \star} \tt \:Coordinate \: of \: A = ( x_{1}, y_{1}) \\ \\\green{ \star} \tt \:Coordinate \: of \: B= ( x_{2}, y_{2}) \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies x = \frac{m x_{2} + nx_{1} }{m + n } \\ \\ \tt: \implies x = \frac{(a - b)3 + (a + b)2}{3 + 2} \\ \\ \tt: \implies x = \frac{3a - 3b + 2a + 2b}{5} \\ \\ \green{\tt: \implies x = \frac{5a - b}{5}} \\ \\ \bold{similarly : } \\ \tt: \implies y = \frac{m y_{2} + ny_{1}}{m + n} \\ \\ \tt: \implies y = \frac{(a + b)3 + (a - b)2}{3 + 2} \\ \\ \tt: \implies y = \frac{3a + 3b + 2a - 2b}{5} \\ \\ \green{\tt: \implies y = \frac{5a + b}{5} } \\ \\ \green{\tt \therefore Coordinate \: of \: O \: is \: \bigg( \frac{5a - b}{5} ,\frac{5a + b}{5} \bigg)}$