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step by step solution and graph each answer

Let's solve the problems one by one.

### 1. Determine the points of intersection of the circle \(x^2 + y^2 - 4x + 2y - 20 = 0\) with the following lines:
   - \(1.3x + 7y - 20 = 0\)
   - \(2.8x - 3y - 7 = 0\)

#### Step 1: Rewrite the Circle Equation in Standard Form
First, we complete the square for the circle's equation:
\[
x^2 - 4x + y^2 + 2y = 20
\]
\[
(x - 2)^2 - 4 + (y + 1)^2 - 1 = 20
\]
\[
(x - 2)^2 + (y + 1)^2 = 25
\]
The circle is centered at \((2, -1)\) with radius \(5\).

#### Step 2: Intersection with \(1.3x + 7y - 20 = 0\)
Solve this system:
\[
1.3x + 7y = 20 \quad \text{(Equation of the line)}
\]
Substitute \(y = \frac{20 - 1.3x}{7}\) into the circle's equation:
\[
(x - 2)^2 + \left(\frac{20 - 1.3x}{7} + 1\right)^2 = 25
\]
This is a quadratic equation in \(x\). Solve for \(x\) and substitute back to find \(y\).

#### Step 3: Intersection with \(2.8x - 3y - 7 = 0\)
Similarly, solve this system:
\[
2.8x - 3y = 7 \quad \text{(Equation of the line)}
\]
Substitute \(y = \frac{2.8x - 7}{3}\) into the circle's equation and solve for \(x\) and \(y\).

### 2. Find the equation of the circle that passes through

Explore the step-by-step solution to find the points of intersection between a circle and two lines. This problem involves solving circle equations and line equations, suitable for advanced high school students and those interested in geometry and algebra.

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