Question

Two sides of a rhombus $ABCD$ parallel to the lines $y=x+2$ and $y=7x+3$. If the diagonals of the rhombus intersect at the point $P(1,2)$ and if the vertex $A$ is on the y-axis find the possible co-ordinates of $A$.

Answer 1

$A$ being on y-axis may be chosen as $(0,a)$. The diagonals intersect at $P(1,2)$. Again we know that diagonals will be parallel to the bisectors of two sides
$\frac{x-y+2}{\sqrt{2}}=\pm \frac{7x-y+3}{5\sqrt{2}}$
$d_1$ is parallel to $x+2y-\frac{7}{2}=0$ and $d_2$ parallel to $2x-y+\frac{13}{6}=0$. The vertex $A$ could be on any of the two diagonals
Hence slope of $AP=2$ or $-1/2$.
$\therefore \frac{2-a}{1-0}=2-\frac{1}{2}\therefore a=0,\frac{5}{2}$
$A$ is $(0,0)$ or $(0,5/2)$