1. Interpret the question

Since vertex $A$ is on the $y$-axis, let $A=(0,a), \quad a\neq 0.$

The sides of the rhombus are parallel to the lines $x-y+2=0 \quad \text{and} \quad 7x-y+3=0.$ So the two side directions have slopes:

• from $x-y+2=0 \Rightarrow y=x+2$, slope $1$
• from $7x-y+3=0 \Rightarrow y=7x+3$, slope $7$

Thus, from $A$, the two adjacent sides of the rhombus go along directions with slopes $1$ and $7$.

2. Take side vectors along these directions

Let the two side vectors from $A$ be $\vec{AB}=t(1,1), \qquad \vec{AD}=s(1,7).$

Because $ABCD$ is a rhombus, adjacent sides are equal: $|\vec{AB}|=|\vec{AD}|.$

Now, $|(1,1)|=\sqrt{2}, \qquad |(1,7)|=\sqrt{50}=5\sqrt{2}.$ So, $|t|\sqrt{2}=|s|\cdot 5\sqrt{2} \Rightarrow |t|=5|s|.$

Hence we may take $t=5k, \qquad s=k$ for some real $k$ (sign can be absorbed into $k$).

Therefore, $\vec{AB}=5k(1,1), \qquad \vec{AD}=k(1,7).$

3. Use the midpoint of diagonals

In a rhombus (parallelogram), diagonals bisect each other. Since they intersect at $P=(1,2),$ we have $P=\frac{A+C}{2}.$

Also, for a parallelogram, $C=A+\vec{AB}+\vec{AD}.$ So, $P=A+\frac{\vec{AB}+\vec{AD}}{2}.$

Now, $\vec{AB}+\vec{AD}=5k(1,1)+k(1,7)=(6k,12k).$ Thus, $P=(0,a)+\frac{(6k,12k)}{2}=(0,a)+(3k,6k)=(3k,a+6k).$

Given $P=(1,2)$, compare coordinates: $3k=1 \Rightarrow k=\frac13.$ Then, $a+6\left(\frac13\right)=2 \Rightarrow a+2=2 \Rightarrow a=0.$ This gives $A=(0,0)$, but the question says $A$ is different from the origin.

So we must also allow the second side direction to be taken in the opposite sense: $\vec{AD}=-k(1,7), \qquad \vec{AB}=5k(1,1).$ Then $\vec{AB}+\vec{AD}=5k(1,1)-k(1,7)=(4k,-2k).$ Hence $P=(0,a)+\frac{(4k,-2k)}{2}=(0,a)+(2k,-k)=(2k,a-k).$

Now use $P=(1,2)$: $2k=1 \Rightarrow k=\frac12,$ $a-k=2 \Rightarrow a=2+\frac12=\frac52.$

Therefore, $A=(0,\tfrac52).$

So the coordinate asked is $\boxed{\frac52}.$

4. Check options

The $y$-coordinate of $A$ is $\dfrac52$, which matches Option A.

5. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

They agree.