A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$.
Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola.

[b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.
1. **Identify the given elements and their relationships:**
   - Circle \( C_1 \) with radius \( r_1 \).
   - Circle \( C_2 \) with radius \( r_2 \).
   - A variable tangent \( t \) to \( C_1 \) intersects \( C_2 \) at points \( A \) and \( B \).
   - Tangents to \( C_2 \) at \( A \) and \( B \) intersect at point \( P \).

2. **Determine the locus of midpoints \( M \) of the chords \( AB \):**
   - The locus of midpoints \( M \) of the chords \( AB \) is the pedal curve of \( C_1 \) with respect to \( O_2 \) (the center of \( C_2 \)).
   - This pedal curve is a Pascal's limaçon \( \mathcal{L} \) with cusp at \( O_2 \) and embracing \( C_1 \).

3. **Relate the distances involving \( M \) and \( P \):**
   - Since \( \overline{O_2M} \cdot \overline{O_2P} = r_2^2 \), it follows that the locus of points \( P \) is the inverse of \( \mathcal{L} \) under the inversion with respect to \( C_2 \).

4. **Determine the nature of the conic \( \mathcal{H} \):**
   - The locus of \( P \) is a conic \( \mathcal{H} \) with focus at \( O_2 \) and focal axis along \( O_1O_2 \) (where \( O_1 \) is the center of \( C_1 \)).
   - For \( \mathcal{H} \) to be an equilateral hyperbola, its eccentricity \( e \) must equal \( \sqrt{2} \).

5. **Set up the polar coordinate system:**
   - Take \( O_2 \) as the pole of the polar coordinate system \((r, \theta)\) and \( O_2O_1 \) as the polar axis (\(\theta = 0^\circ\)).
   - Thus, \( O_1 \equiv (d, 0) \), where \( d \) is the distance between the centers of \( C_1 \) and \( C_2 \).

6. **Write the equations of \( \mathcal{L} \) and \( \mathcal{H} \):**
   - The equation of the Pascal's limaçon \( \mathcal{L} \) is:
     \[
     r = r_1 + d \cos \theta
     \]
   - The equation of the conic \( \mathcal{H} \) (inverse of \( \mathcal{L} \) under inversion with respect to \( C_2 \)) is:
     \[
     r = \frac{r_2^2}{r_1 + d \cos \theta}
     \]

7. **Determine the condition for \( \mathcal{H} \) to be an equilateral hyperbola:**
   - For \( \mathcal{H} \) to be an equilateral hyperbola, its eccentricity \( e \) must be \( \sqrt{2} \).
   - The eccentricity \( e \) of the hyperbola is given by:
     \[
     e = \frac{d}{r_1}
     \]
   - Setting \( e = \sqrt{2} \), we get:
     \[
     \sqrt{2} = \frac{d}{r_1}
     \]
   - Solving for \( d \), we find:
     \[
     d = r_1 \sqrt{2}
     \]

The final answer is \(\boxed{d = r_1 \sqrt{2}}\)