Concentric with x²+20x+y²-14y+145=0 diameter 12

Let's solve the problem of finding the equation of a circle concentric with the given circle \(x^2 + 20x + y^2 - 14y + 145 = 0\) and having a diameter of 12.

### Step 1: Find the center of the given circle
The equation of the given circle is:

\[
x^2 + 20x + y^2 - 14y + 145 = 0
\]

To find the center, we need to complete the square for both \(x\) and \(y\).

**For \(x\):**

\[
x^2 + 20x = (x + 10)^2 - 100
\]

**For \(y\):**

\[
y^2 - 14y = (y - 7)^2 - 49
\]

Substitute these back into the equation:

\[
(x + 10)^2 - 100 + (y - 7)^2 - 49 + 145 = 0
\]

Simplify:

\[
(x + 10)^2 + (y - 7)^2 - 4 = 0
\]

\[
(x + 10)^2 + (y - 7)^2 = 4
\]

So, the center of the given circle is \((-10, 7)\), and its radius is \(r = 2\).

### Step 2: Calculate the radius of the new circle
The new circle is concentric with the given one, meaning it has the same center \((-10, 7)\). The diameter of the new circle is given as \(12\), so the radius \(r\) of the new circle is:

\[
r = \frac{12}{2} = 6
\]

### Step 3: Write the equation of the new circle
The general form of a circle's equation is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting the center \((-10, 7)\) and the radius \(r = 6\):

\[
(x + 10)^2 + (y - 7)^2 = 6^2
\]

\[
(x + 10)^2 + (y - 7)^2 = 36
\]

### Final Answer:
The equation of the circle concentric with the given circle and having a diameter of 12 is:

\[
(x + 10)^2 + (y - 7)^2 = 36
\]

Would you like further details or have any questions? Here are some related problems to consider:

1. Find the equation of a circle with the same center but a radius of 8.
2. Determine the area of the circle with the equation \((x + 10)^2 + (y - 7)^2 = 64\).
3. What is the equation of a circle concentric with the given circle but passing through the point \((0, 0)\)?
4. Calculate the circumference of the new circle with a diameter of 12.
5. What is the standard form of the equation of a circle?
6. How do you find the distance between two concentric circles?
7. Derive the equation of a circle with center \((5, -3)\) and radius \(7\).
8. What is the equation of a circle passing through \((0, 0)\) with center \((2, -1)\)?

**Tip:** When working with circles, remember that concentric circles share the same center but differ in radius.

Learn how to find the equation of a circle concentric with x² + 20x + y² - 14y + 145 = 0 and having a diameter of 12. This problem involves circle geometry and requires knowledge of the standard form of the equation of a circle. Suitable for high school students in Grades 10-12.

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