To solve this question effectively, I will first extract the key mathematical problem from the image. Once I've processed the relevant details, I’ll provide a clear solution and explanation. Allow me a moment to analyze this content.The question involves a point $P(-x, y)$ on a specific triangle's circumcircle in the second quadrant. It also mentions point $A \left( \frac{1}{x}, \alpha \right)$ and asks us to determine the value of $\alpha$ if $A$ also lies on the same triangle's circumcircle.

Steps to Solve:

1. Properties of a Circumcircle:

   • The circumcircle is the unique circle passing through all three vertices of the triangle. Any point $(x, y)$ lying on it satisfies its equation.

2. The Equation of the Circle: Let the equation of the circumcircle be of the general form: $x^2 + y^2 = r^2$ Since $P(-x, y)$ is given as on the circumcircle, substitute its coordinates: $(-x)^2 + y^2 = r^2 \implies x^2 + y^2 = r^2$

3. Point $A \left( \frac{1}{x}, \alpha \right)$ lies on the Circle: Substitute $A \left( \frac{1}{x}, \alpha \right)$ into the equation of the circle: $\left( \frac{1}{x} \right)^2 + \alpha^2 = r^2$ Since we know