If $ \mathrm{P} \frac{a}{3}, 4 $ is the mid-point of the line segment joining the points $ \mathrm{Q}(-6,5) $ and $ \mathrm{R}(-2,3) $, then the value of $ a $ is
(A) $ -4 $
(B) $ -12 $
(C) 12
(D) $ -6 $

Given:

A line segment joining the points $Q( -6,\ 5)$ and$R( -2,\ 3)$ and its mid-point is $P\left(\frac{a}{2} ,\ 4\right)$

To do:

We have to find the value of $a$.

Solution:

If there is a line segment joining two points$ ( x_{1} ,\ y_{1} )$ and $( x_{2} ,\ y_{2})$,

Then its mid-point $( x,\ y) =\left(\frac{x_{1} +x_{2}}{2} ,\ \frac{y_{1} +y_{2}}{2}\right)$

Therefore,

$P\left(\frac{a}{2} ,4\right) =\left(\frac{-6-2}{2} ,\frac{5+3}{2}\right)$

$\Rightarrow \ P\left(\frac{a}{2} ,4\right) =\left(\frac{-8}{2} ,\frac{8}{2}\right)$

$\Rightarrow \ P\left(\frac{a}{2} ,4\right) =\left(\frac{-8}{2} ,4\right)$

$\Rightarrow \frac{a}{2} =\frac{-8}{2}$

$\Rightarrow a=-4$

The value of $a$ is $-4$.

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Updated on: 10-Oct-2022

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