To determine if two ellipses intersect, we need to analyze the geometric relationships between their equations. The equations given describe two ellipses:

1. $\frac{(x - x_1)^2}{a_1^2} + \frac{(y - y_1)^2}{b_1^2} = 1$
2. $\frac{(x - x_2)^2}{a_2^2} + \frac{(y - y_2)^2}{b_2^2} = 1$

To find the points of intersection (if any), we must simultaneously solve these two equations for $x$ and $y$. However, solving this explicitly is difficult, and instead, we can approach it using a geometric condition.

Step 1: Set up a system of equations

The two equations are:

$\frac{(x - x_1)^2}{a_1^2} + \frac{(y - y_1)^2}{b_1^2} = 1 \quad \text{(Equation 1)}$ $\frac{(x - x_2)^2}{a_2^2} + \frac{(y - y_2)^2}{b_2^2} = 1 \quad \text{(Equation 2)}$

We need to find the values of $x$ and $y$ that satisfy both of these equations simultaneously.

Step 2: Find a parameterized form (Optional)

Since solving the system directly can be difficult, we often look at parametric forms for ellipses. However, for our case, we will consider a simpler approach using a derived condition from the ellipses' geometric properties.

Step 3: Geometric Interpretation & Formulation

To find whether the ellipses intersect, we need to check the relative distances between their centers $(x_1, y_1)$ and $(x_2, y_2)$, and how the radii of the ellipses relate to each other. The general condition for the intersection of two ellipses is based on their distance and relative scaling.

Distance Between Centers

Let the distance between the centers of the two ellipses be $d$:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Effective Radii

For each ellipse, the effective radii depend on both the axes $a_1, b_1$ and $a_2, b_2$. To generalize the situation for comparison, we define a normalized effective radius as:

1. For ellipse 1, the maximum effective radius is given by $R_1 = \sqrt{a_1^2 + b_1^2}$.
2. For ellipse 2, the maximum effective radius is $R_2 = \sqrt{a_2^2 + b_2^2}$.

Condition for Intersection

To determine whether the ellipses intersect, we need to check whether the distance between their centers $d$ is smaller than the sum of their effective radii, but also not too small such that one ellipse is inside the other. The condition for intersection is:

$|R_1 - R_2| \leq d \leq R_1 + R_2$

This inequality ensures that the ellipses are neither too far apart nor one entirely inside the other. If $d$ is within this range, the ellipses intersect.

Step 4: Formula for Intersection

We can express this condition as a function $f(k)$, where $k$ is a variable related to the relative geometry. Let $k$ be defined as:

$k = \frac{d}{\sqrt{a_1^2 + b_1^2} + \sqrt{a_2^2 + b_2^2}}$

This scales $d$ relative to the sum of the effective radii of the two ellipses. The ellipses will intersect if:

$0 < k \leq 1$

• If $k = 0$, the ellipses do not intersect (they are too far apart or one is inside the other).
• If $0 < k \leq 1$, the ellipses intersect.

Step 5: Conclusion

Thus, the ellipses will intersect if the following condition holds:

$\left| \sqrt{a_1^2 + b_1^2} - \sqrt{a_2^2 + b_2^2} \right| \leq \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \leq \sqrt{a_1^2 + b_1^2} + \sqrt{a_2^2 + b_2^2}$

Or, equivalently, the ellipses intersect if:

$0 < k \leq 1$

If $k > 1$, they do not intersect.

Do you need any further explanation or details on how to implement this approach numerically?