Intersecting Circles Distance Calculation

Sure, let's solve this problem step by step.

1. Define the circles: We can define the two circles with their centers and radii. Let's place the smaller circle at the origin of our coordinate system. So, the equation of the smaller circle (Circle 1) with radius 1 is $x^2 + y^2 = 1$. The larger circle (Circle 2) with radius 2 passes through the center of the smaller circle, so its center is at (2, 0). Therefore, the equation of the larger circle is $(x-2)^2 + y^2 = 4$.

2. Find the intersection points: To find the intersection points of the two circles, we need to solve the two equations simultaneously. This will give us the coordinates of the points where the circles intersect.

3. Calculate the distance between the intersection points: Once we have the coordinates of the intersection points, we can use the distance formula to find the distance between them.

Let's start by finding the intersection points. We can do this by setting the two equations equal to each other and solving for $x$ and $y$.

Let's proceed with these calculations.

The solutions to the system of equations are $(x, y) = \left(\frac{1}{4}, -\frac{\sqrt{15}}{4}\right)$ and $(x, y) = \left(\frac{1}{4}, \frac{\sqrt{15}}{4}\right)$. These are the coordinates of the points where the two circles intersect.

Now, let's calculate the distance between these two points using the distance formula:

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a plane is given by

Substituting the coordinates of our points into this formula, we get:

Let's compute this.

The exact distance between the two points at which the circles intersect is $\frac{\sqrt{15}}{2}$.