The graph $y = x^2$ meets the line $y = k$ (where $k > 0$) at points $P$ and $Q$ as shown in the diagram - HSC - SSCE Mathematics Advanced - Question 10 - 2023 - Paper 1

To find the length of the interval $ST$ where the graph $y = \frac{x^2}{a^2}$ intersects the line $y = k$, we first express $y = k$ in terms of $x$:

1. Set the equations equal to each other: $k = \frac{x^2}{a^2}$

2. Rearranging gives us: $x^2 = ak$ Taking the square root: $x = \pm \sqrt{ak}$

3. Thus, the points of intersection, $S$ and $T$, occur at $x = \sqrt{ak}$ and $x = -\sqrt{ak}$. The length of $ST$ can be found by calculating the distance between these two points: $|x_{T} - x_{S}| = |-\sqrt{ak} - \sqrt{ak}| = 2\sqrt{ak}$

4. We already know from the previous graph that the length of $PQ = L$, which is computed as: $L = 2\sqrt{a} \sqrt{k} = 2\sqrt{ak}$

5. Therefore, since the relationship is proportional by a factor of $a$, the new length $ST$ becomes: $ST = aL$

In conclusion, the length of $ST$ is given by: $\text{Answer: } aL$