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The midpoint of the line segment joining $(2 a, 4)$ and $(- 2, 2 b)$ is $(1, 2 a + 1)$ . The values of $a$ and $b$ are

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a = b, b = - 1

a = 2, b = - 3

a = 3, b = - 2

a = 2, b = 3

answer is D.

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Detailed Solution

It is given that, the midpoint of the line segment joining

(2 a, 4)

and

(- 2, 2 b)

(1, 2 a + 1)

.
Now midpoint of two points (

x_{1}, y_{1})

and (

x_{2}, y_{2}) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Given, midpoint of

(2 a, 4)

and

(- 2, 2 b)

will be,

(1, 2 a + 1)

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

Now equating abscissa and ordinate we get,

\frac{2 a - 2}{2} = 1

\Rightarrow 2 a - 2 = 2

\Rightarrow 2 a = 4

\Rightarrow a = \frac{4}{2}

\Rightarrow a = 2

….(1)
Also,

\frac{4 + 2 b}{2} = 2 a + 1

\Rightarrow 4 + 2 b = 2 (2 a + 1)

\Rightarrow 4 + 2 b = 4 a + 2

Now putting the value of a in the above expression we get,

\Rightarrow 4 + 2 b = 4 \times 2 + 2

\Rightarrow 2 b + 4 = 10

\Rightarrow 2 b = 10 - 4

\Rightarrow 2 b = 6

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

\Rightarrow b = 3

So,

a = 2, b = 3

. So the correct option is 4.

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