Answer
${{P}_{2}}=\left( 1,2 \right)$
Work Step by Step
Here,
${{P}_{1}}=\left( {{x}_{1}},{{y}_{1}} \right)=\left( -3,6 \right)$ and ${{P}_{2}}=\left( {{x}_{2}},{{y}_{2}} \right)$
By using midpoint formula, the coordinate of midpoint of two points ${{P}_{1}}=\left( -3,6 \right)$ and${{P}_{2}}=\left( {{x}_{2}},{{y}_{2}} \right)$ is $M=(x,y)$, where $x=\frac{-3+{{x}_{2}}}{2}$ and $\,y=\frac{6+{{y}_{2}}}{2}$
The midpoint of the line segment joining the points ${{P}_{1}}$ and ${{P}_{2}}$ is $\left( -1,4 \right)$.
Therefore, $x=-1$and $y=4$
$\Rightarrow \,\,-1=\frac{-3+{{x}_{2}}}{2}$ and $4=\frac{6+{{y}_{2}}}{2}$
Solving each equation gives:
$-1=\dfrac{-3+{{x}_{2}}}{2}$
$ -2={{x}_{2}}-3$
$ -2+3={{x}_{2}}-3+3$
$ 1={{x}_{2}}$
and
$4=\frac{6+{{y}_{2}}}{2}$
$\,\,8\,={{y}_{2}}+6$
$\,\,8-6\,={{y}_{2}}+6-6$
$\,\,2\,={{y}_{2}}$
${{y}_{2}}=2$
Thus,
${{P}_{2}}=\left( 1,2 \right)$
