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

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.

Solution:

The points are A(2a, 4) and B(−2, 3b).
Let C(1, 2a + 1) be the mid point of AB. Then:

$x=\frac{x_{1}+x_{2}}{2}, y=\frac{y_{1}+y_{2}}{2}$

$\Rightarrow 1=\frac{2 a+(-2)}{2}, 2 a+1=\frac{4+3 b}{2}$

$\Rightarrow 2=2 a-2,4 a+2=4+3 b$

$\Rightarrow 2 a=2+2,4 a-3 b=4-2$

$\Rightarrow a=\frac{4}{2}, 4 a-3 b=2$

$\Rightarrow a=2,4 a-3 b=2$

Putting the value of $a$ in the equation $4 a+3 b=2$, we get:

$4(2)-3 b=2$

$\Rightarrow-3 b=2-8=-6$

$\Rightarrow b=\frac{6}{3}=2$

Therefore, $a=2$ and $b=2$.

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