Step 1: Parametric point and tangent of the hyperbola.
For $x^2-y^2=a^2$: point $(a\sec\theta, a\tan\theta)$ and tangent $x\sec\theta-y\tan\theta=a$.
Step 2: Set up tangents at two ends of a normal chord.
Let the chord join parameters $\theta$ and $\phi$. The tangents $x\sec\theta-y\tan\theta=a$ and $x\sec\phi-y\tan\phi=a$ meet at $(h,k)$.
Step 3: Solve the pair of tangent equations.
From the two equations: $h\sec\theta-k\tan\theta=a$ and $h\sec\phi-k\tan\phi=a$, we can express $h$ and $k$ in terms of $\sec\theta,\tan\theta,\sec\phi,\tan\phi$.
Step 4: Impose the normal chord condition.
For $x^2-y^2=a^2$, two points are connected by a normal if a specific relation holds between their parameters $\theta$ and $\phi$. Applying this condition allows us to eliminate the parameters.
Step 5: Derive the locus.
After elimination of parameters, the locus of the intersection point $(h,k)$ satisfies $a^2(k^2-h^2)=4h^2k^2$. Writing in $(x,y)$: \[a^2(y^2-x^2)=4x^2y^2.\]
Step 6: State the answer.
\[ \boxed{a^2(y^2-x^2)=4x^2y^2} \]