The coordinates of the midpoint of the line segment joining the points $(2 a + 2, 3)$ and $(4, 2 b + 1)$ are $(2 a, 2 b)$ . Find the values of $a$ and $b$ .

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Solution

Given, $(2 a, 2 b)$ is the midpoint of $(2 a + 2, 3), (4, 2 b + 1)$
Therefore, $(\frac{2 a + 2 + 4}{2}, \frac{3 + 2 b + 1}{2}) = (2 a, 2 b)$
$\Rightarrow (\frac{2 a + 6}{2}, \frac{2 b + 4}{2}) = (2 a, 2 b)$
$\Rightarrow \frac{2 a + 6}{2} = 2 a$ and $\frac{2 b + 4}{2} = 2 b$
$\Rightarrow 2 a + 6 = 4 a$ and $2 b + 4 = 4 b$
$\Rightarrow 2 a = 6$ and $2 b = 4$
$\Rightarrow a = 3$ and $b = 2$

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