Question

(-1, 1) are the co-ordinates of the mid point of line AB joining the points A(-3,b) and B(1,b+4), then the value of b is​

Answer 1

Given:

(-1, 1) are the co-ordinates of the mid point of line segment AB joining the points A(-3,b) and B(1,b+4)

To Find:

• Value of b

Mid-Point Formula

$\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){ P }\put(11.7,5.9){ Q }\put(5.4,5.5){ (x_1\,,\,y_1) }\put(11.4,5.5){ (x_2\,,\,y_2) }\put(8.5,6){\circle*{0.2}}\put(8.3,6.3){ R }\put(8,5.5){ (x\,,\,y) }\put(7.2,6.3){ 1 }\put(9.9,6.3){ 1 }\put(11.7,5.9){ Q }\end{picture}$

If R divides the line segment joining points P and Q, internally in the ratio 1 : 1 .i.e., if R is the mid-point of PQ , then

$\sf{(x,y)=\bigg(\dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2}\bigg)}$

Solution:

Let M be the mid-point of AB

$\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){ A }\put(11.7,5.9){ B }\put(5.4,5.5){ (-3\,,\,b) }\put(11.4,5.5){ (1\,,\,b+4) }\put(8.5,6){\circle*{0.2}}\put(8.3,6.3){ M }\put(8,5.5){ (-1\,,\,1) }\put(7.2,6.3){ 1 }\put(9.9,6.3){ 1 }\put(11.7,5.9){ B }\end{picture}$

On applying Mid-Point Formula, we get

$\longrightarrow\sf{(-1,1)=\bigg(\dfrac{-3+1}{2},\dfrac{b+b+4}{2}\bigg)}$

$\longrightarrow\sf{(-1,1)=\bigg(\dfrac{-2}{\: 2},\dfrac{2b+4}{2}\bigg)}$

$\longrightarrow\sf{(-1,1)=\bigg(\dfrac{-\cancel{2}}{\: \cancel{2}},\dfrac{\cancel{2}(b+2)}{\cancel{2}}\bigg)}$

$\longrightarrow\sf{(-1,1)=\big(-1,b+2\big)}$

On comparing y-coordinate of both the sides, we get

⇒ 1= b+2

⇒ b+2= 1

⇒ b= 1-2

⇒ b= -1

Hence, the value of b is -1.