1. Find the center and radius of the circle 9x ^ 2 + 36x + 9y ^ 2 + 72y + 155 = 6 Sketch its graph and indicate the center. (7 points

2. the standard equation of the circle that passes through the endpoints of a diameter P (1, - 1) and Q (-5,-9). Sketch the graph of the circle and indicate the center. (7 points)

3. Find the standard equation of a circle with the center at (-5,-1) and tangent to the line x + y + 10 = 0 (7 points)

4. The circle passes through the points P(1,- 2) Q(5, 4) and R(10, 5) Write the equation of the circle in a general form. (8 points)

5. The circle with the center at (-2, 3) is tangent to the x-axis. Write the equation of the circle in standard form. (6 points)

Let's work through each problem step by step:

### 1. Find the center and radius of the circle and sketch its graph.

Given the equation:

\[
9x^2 + 36x + 9y^2 + 72y + 155 = 6
\]

First, we simplify this equation:

\[
9(x^2 + 4x) + 9(y^2 + 8y) + 155 = 6
\]

Subtract 155 from both sides:

\[
9(x^2 + 4x) + 9(y^2 + 8y) = -149
\]

Now, divide the entire equation by 9:

\[
x^2 + 4x + y^2 + 8y = -\frac{149}{9}
\]

Next, complete the square for both \(x\) and \(y\):

- For \(x\), take half of 4 and square it: \( (4/2)^2 = 4 \).
- For \(y\), take half of 8 and square it: \( (8/2)^2 = 16 \).

So, add and subtract these values inside the equation:

\[
(x^2 + 4x + 4) + (y^2 + 8y + 16) = -\frac{149}{9} + 4 + 16
\]

\[
(x + 2)^2 + (y + 4)^2 = \frac{-149 + 180}{9} = \frac{31}{9}
\]

Now, we have the equation of the circle in standard form:

\

Learn how to solve circle equations step-by-step with detailed explanations and examples. Includes finding the center and radius, standard form derivation, and tangent line applications. Suitable for high school students.

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