Constructing a circle of given radius in a pencil defined by a circle and a radical axis

In this exercise you are given a circle centred at \(O_1\) and passing through the point \(P\). The centre \(O_1\) lies on the line \(AB\). You are also given a radical axis (coloured in red), which is perpendicular to the line \(AB\), and a line segment \(CD\). Your task is to construct a circle with centre on the line \(AB\) and radius equal to \(|CD|\) such that the red line is the radical axis of the two circles. Equivalently, you need to construct a circle with radius \(|CD|\) which is in the pencil defined by the circle centred at \(O_1\) and the red radical axis.

Hint. You should read through the different properties of the radical axis and the power of a point and see which one is the most useful here. You may find the previous exercise very useful.

The diagram below contains an example of what the solution should look like. (There are actually two circles with the required property; the diagram below only shows one of them.) You can see how the circle changes when you change the length of \(|CD|\), change the position of the radical axis by moving \(X\) along the line \(AB\), or change the original circle by moving the points \(O_1\) and \(P\).