Question

The diagonals of a rhombus $ABCD$ intersect at the point $(1,2)$ and its two sides are parallel to the lines $x-y+2=0$ and $7x-y+3=0$. if the vertex $A$ be situated on y-axis, find the possible coordinates of $A$.

Answer 1

Choose the vertex $A$ to be $(0,k)$ as it is on y-axis
The sides through $A$ being parallel to given lines and pass through $(0,k)$
$AB,x-y+k=0$
$AD,7x-y+k=0$
The point $P$ which is intersection of diagonals will be on the bisectors of these sides
$\frac{x-y+k}{\sqrt{2}}=\pm \frac{7x-y+k}{5\sqrt{2}}$
They pass through $(1,2)$
$\therefore$ $k=(0,5/2)$. Hence $A$ is $(0,0)$ or $(0,5/2)$