Question

Let the line segment joining $A(6,3)$ and $B(-1,-4)$ is doubled in length by adding equal segments to both the ends. If $P(x_1,y_1)$ and $Q(x_2,y_2)$ are the new end points, then the value of $5(x_1+y_2)+4(x_2+y_1)$ is

Answer 1

Let $M$ be the mid-point of $AB$.
$PA=AM=MB=BQ(\because PQ=2AB)$
$M=(\frac{5}{2},-\frac{1}{2})$
$A$ is the mid-point of $PM$
$\Rightarrow P=2A-M=(\frac{19}{2},\frac{13}{2})=(x_1,y_1)$
$B$ is the mid-point of $MQ$
$\Rightarrow Q=2B-M=(-\frac{9}{2},-\frac{15}{2})=(x_2,y_2)$

$\therefore 5(x_1+y_2)+4(x_2+y_1)$
$=5(\frac{19}{2}-\frac{15}{2})+4(-\frac{9}{2}+\frac{13}{2})$
$=18$