To find the equation of the circle that passes through the point $(1,1)$ and the points of intersection of two given circles, we can follow these steps:

Step 1: Write the given equations of the circles

The first circle is given as: x^2 + y^2 - 6x + 8 = 0 \tag{1} Let's analyze this equation.

The standard form of a circle's equation is: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

Rewriting the given equation (1) in a more recognizable form, we complete the square for both $x$ and $y$ terms:

• For $x$, we have $x^2 - 6x$. To complete the square, add and subtract $\left(\frac{6}{2}\right)^2 = 9$.
• For $y$, we have $y^2$, which is already a perfect square.

Thus, the equation becomes: $(x^2 - 6x + 9) + y^2 = 9 - 8$ or (x - 3)^2 + y^2 = 1 \tag{2} This represents a circle with center $(3, 0)$ and radius $r = 1$.

Step 2: The second circle's equation

If there's another circle involved, it should be provided. However, if the second circle is the same as the first, we proceed with just one equation. If you have the second circle's equation, please provide it for further steps.

Step 3: Points of intersection of the two circles

The points of intersection of two circles can be found by solving the system of equations of both circles. However, if no second equation is given, we assume that you're referring to the points on the first circle.

Step 4: Equation of the new circle

The equation of the circle passing through the point $(1,1)$ and the intersection points of the two given circles can be found using the radical axis method or by applying the general form of the equation of a circle through two fixed points.

This would be written in the form: $S_1(x, y) + \lambda S_2(x, y) = 0$ where $S_1(x, y)$ and $S_2(x, y)$ are the equations of the two given circles, and $\lambda$ is a constant.

We need the second circle's equation to proceed further.

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Would you like to provide the second circle's equation, or do you want me to continue with just the first? Feel free to ask for more details if needed!

Here are some related questions for further exploration:

1. How to find the points of intersection of two circles algebraically?
2. What is the general form of the equation of a circle passing through two points?
3. How do we complete the square for a circle equation?
4. What is the significance of the radical axis in geometry?
5. How can you derive the equation of a circle from its center and radius?

Tip: When working with the intersection of two circles, solving the system of equations can sometimes involve substitution or elimination methods. Keep the terms organized to avoid errors when solving for intersection points!