question_answer

If \[P\left( \frac{a}{3},4 \right)\]is the mid-point of the line segment joining the points \[Q\left( -6,\,5 \right)\]and\[R\left( -2,\,3 \right)\], then the value of a is

A) -4

B) -12

C) 12

D) -6

Correct Answer: B

Solution :

Given that, \[P\left( \frac{a}{3},\,4 \right)\]is the mid-point of the line segment joining the points Q (-6, 5) and R (-2, 3), which shows in the figure given below

\[\therefore\] Mid-point of \[QR=P\left( \frac{-6-2}{2},\,\frac{5+3}{2} \right)\]

=P(-4,4)

[since, mid-point of line segment having points \[\left( {{x}_{1}},\,{{y}_{1}} \right)\] and \[\left( {{x}_{2}},\,{{y}_{2}} \right)\]

\[\left. =\left( \frac{\left( {{x}_{1}}+{{x}_{2}} \right)}{2},\frac{\left( {{y}_{1}}+{{y}_{2}} \right)}{2} \right) \right]\]

But mid-point \[P\left( \frac{a}{3},\,4 \right)\] is given.

\[\therefore \,\,\,\left( \frac{a}{3},\,4 \right)=\left( -4,\,\,4 \right)\]

On comparing the coordinates, we get

\[\frac{a}{3}=-4\]

\[\Rightarrow \,\,\,a=-12\]

Hence, the required value of a is -12.