Given that y=ax^2 is the similar transformation of y=x^2, determine the ratio of similitude.

In my solution book, I follow every step, but not the answer.

When \left\{ \begin{gathered}x'=kx \\ y'=ky \end{gathered} \right., \left\{ \begin{gathered}x=\frac{1}{k}x' \\ y=\frac{1}{k}y' \end{gathered} \right. (2)

Substituting (2) into y=x^2

\frac{1}{k}y'=\bigg ( \frac{1}{k}x' \bigg )^2
y'=\frac {1}{k}(x')^2
\therefore k=\frac {1}{a}

How do you get k=\frac {1}{a}