Question
Two sides of a rhombus $ABCD$ are parallel to lines $y = x + 2$ and $y = 7x + 3$. If the diagonals of the rhombus intersect at point $(1,2)$ and the vertex $A$ is on the $y$-axis is, then the possible coordinates of $A$ are:
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Solution
Sol. [Key Concept: Each diagonal of the rhombus is parallel to an angle-bisector of the two side directions and passes through the centre (1,2).]
The diagonal of rhombus is parallel to angle bisector of given lines
$${\frac{y - x - 2}{\sqrt{2}} = \pm \frac{y - 7x - 3}{5\sqrt{2}} }{\Rightarrow 4y + 2x - 7 = 0,6y + 12x - 13 = 0 }$$∴ Diagonals are $2y + x = 5$
and $\ 2x - y = 0$
∴ Possible coordinates of $A$ are $\left( 0,\frac{5}{2} \right)$ or $(0,0)$
✔ (a) $(0,\frac{52)}{}$ lies on one diagonal through $(1,2)$.
✔ (b) $(0,0)$ lies on the other diagonal through $(1,2)$.
✘ (c) $(0,5)$ is on neither diagonal.
✘ (d) $(0,3)$ is on neither diagonal.
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