Question

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

• A. $(0,0)$
• B. $(0,5)$
• C. $(0,1)$
• D. $(0,5/2)$

Answer 1

The correct option is A $(0,0)$

Since the required coordinates is on $y-$ axis it may be chosen as $(0,a)$. Now diagonals intersects at $(1,2)$ we know diagonals will be parallel to the angle bisector of the two sides $y=x+2$ and $y=7x+3$

$i.e\frac{x-y+2}{\sqrt{2}}=\pm \frac{7x-y+3}{5\sqrt{2}}$

$\Rightarrow 2x+4y-7$ and $12x-6y+13=0$

Slope $=\frac{-1}{2}$ Slope $=2$

Let diagonal of be parallel to $2x+4y-7=0$

diagonal $D$ be parallel to $12x-6y+13=0$

The vertex $A$ could be on any of the two diagonals Hence slope of $AP$ is either $\frac{-1}{2}$ or $2$

$\Rightarrow \frac{2-a}{1-0}=2$ if $\frac{2-a}{1-0}=\frac{-1}{2}$

then $a=0$ then $a=\frac{5}{2}$

Coordinates are $(0,0)$ or $(0,\frac{5}{2})$