Question

The midpoint of the line segment joining A(2a, 4) and B(−2, 3b) is C(1, 2a + 1). Find the values of a and b.

Answer 1

The points are A(2a, 4) and B(−2, 3b).
Let C(1, 2a + 1) be the mid point of AB. Then:
$$\begin{gathered} x=\frac{x_1+x_2}{2},y=\frac{y_1+y_2}{2} \\ \Rightarrow 1=\frac{2a+\left(-2\right)}{2},2a+1=\frac{4+3b}{2} \\ \Rightarrow 2=2a-2,4a+2=4+3b \\ \Rightarrow 2a=2+2,4a-3b=4-2 \\ \Rightarrow a=\frac{4}{2},4a-3b=2 \\ \Rightarrow a=2,4a-3b=2 \\ \text{Putting}\mathrm{the}\mathrm{value}\mathrm{of}a\mathrm{in}\mathrm{the}\mathrm{equation}4a+3b=2,\text{we get}: \\ 4\left(2\right)-3b=2 \\ \Rightarrow -3b=2-8=-6 \\ \Rightarrow b=\frac{6}{3}=2 \\ \mathrm{Therefore},a=2\mathrm{and}b=2. \end{gathered}$$