15. Reflection in the line $l$ transforms the point with coordinates $(5,3)$ into the point with coordinates $(1,-1)$.
What is the equation of the line $l$ ?
A $y=x-2$
B $y=1$
C $x=3$
D $y=2-x$
E $y=4-x$
15. E The line $l$ is the perpendicular bisector of the line segment joining points $(5,3)$ and $(1,-1)$.
The midpoint of this line segment is $\left(\frac{5+1}{2}, \frac{3+(-1)}{2}\right)=(3,1)$.
The gradient of the line segment is $\frac{3-(-1)}{5-1}=\frac{4}{4}=1$. Hence the gradient of $l$ is -1 .
Let the equation of $l$ be $y=-x+c$. As $l$ passes through the point $(3,1), 1=-3+c$. So $c=4$. Hence the equation of $l$ is $y=-x+4$ or $y=4-x$.