I am working on a geometry problem involving two circles, $O_1$ and $O_2$, with the following properties:

Center-to-center distance: The distance between the centers of $O_1$ and $O_2$ is 5 units. Radii: The radius of $O_1$ is 2 units, and the radius of $O_2$ is 1 unit.

Condition:

There exists a point $P$ which serves as the center of a third circle $O$ such that:

The distance between the two intersection points of circle $O$ and circle $O_1$ is 4 units. The distance between the two intersection points of circle $O$ and circle $O_2$ is 2 units. Problem:

Determine the locus of the point $P$ that satisfies the above condition.

I have derived that the answer is $x = \frac{11}{5}$ by draw. However, I am struggling to understand the steps leading to this result. Could someone provide a detailed explanation or derivation?

I am working on a geometry problem involving two circles, $O_1$ and $O_2$, with the following properties:

Center-to-center distance: The distance between the centers of $O_1$ and $O_2$ is 5 units. Radii: The radius of $O_1$ is 2 units, and the radius of $O_2$ is 1 unit.

Condition:

There exists a point $P$ which serves as the center of a third circle $O$ such that:

The distance between the two intersection points of circle $O$ and circle $O_1$ is 4 units. The distance between the two intersection points of circle $O$ and circle $O_2$ is 2 units. Problem:

Determine the locus of the point $P$ that satisfies the above condition.

I have derived that the answer is $x = \frac{11}{5}$ by draw. However, I am struggling to understand the steps leading to this result. Could someone provide a detailed explanation or derivation?

I am working on a geometry problem involving two circles, $O_1$ and $O_2$, with the following properties:

Center-to-center distance: The distance between the centers of $O_1$ and $O_2$ is 5 units. Radii: The radius of $O_1$ is 2 units, and the radius of $O_2$ is 1 unit. Condition:

There exists a point $P$ which serves as the center of a third circle $O$ such that:

The distance between the two intersection points of circle $O$ and circle $O_1$ is 4 units. The distance between the two intersection points of circle $O$ and circle $O_2$ is 2 units. Problem:

Determine the locus of the point $P$ that satisfies the above condition.

I have derived that the answer is $x = \frac{11}{5}$ by draw. However, I am struggling to understand the steps leading to this result. Could someone provide a detailed explanation or derivation?

I am working on a geometry problem involving two circles, $O_1$ and $O_2$, with the following properties:

Center-to-center distance: The distance between the centers of $O_1$ and $O_2$ is 5 units. Radii: The radius of $O_1$ is 2 units, and the radius of $O_2$ is 1 unit.

Condition:

There exists a point $P$ which serves as the center of a third circle $O$ such that:

The distance between the two intersection points of circle $O$ and circle $O_1$ is 4 units. The distance between the two intersection points of circle $O$ and circle $O_2$ is 2 units. Problem:

Determine the locus of the point $P$ that satisfies the above condition.

I have derived that the answer is $x = \frac{11}{5}$ by draw. However, I am struggling to understand the steps leading to this result. Could someone provide a detailed explanation or derivation?