(a) If the mid-point of the line segment joining the points A(3, 4) and B ( k, 6) is P ( x, y ) and x + y -10=0, find the value of k.

\[\begin{aligned} & \mathrm{P}(\mathrm{x}, \mathrm{y}) \text { is the mid - point } \\ & \therefore(\mathrm{x}, \mathrm{y})=\left(\frac{3+\mathrm{k}}{2}, \frac{4+6}{2}\right) \\ & \mathrm{x}=\frac{3+\mathrm{k}}{2}, \mathrm{y}=5 \\ & \mathrm{x}+\mathrm{y}-10=0 \\ & \frac{3+\mathrm{k}}{2}+5-10=0 \\ & \mathrm{k}=7 \end{aligned} \]

B \(\displaystyle (2,8)\)

Coordinates of mid - point Q of \(\displaystyle \mathrm{AB}=(0,5)\)

Coordinates of mid - point P of \(\displaystyle \mathrm{AQ}=\left(-1, \frac{7}{2}\right)\)

Coordinates of mid - point R of \(\displaystyle \mathrm{BQ}=\left(1, \frac{13}{2}\right)\)