The length of the diameter of the circle `x^2+y^2−4x−6y+4=0`

Options

A. (5,0),(5,-6),(-1,0),(-1,-6)
B. (5,1),(5,-6),(-,1),(-1,-6)
C. (5,-1),(5,6),(-1,0),(1,-6)
D. (0,5),(-6,5),(0,-1),(6,1)

Answer

To find the length of the diameter of the circle given by the equation \(x^2 + y^2 - 4x - 6y + 4 = 0\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the circle:
\[ x^2 + y^2 - 4x - 6y + 4 = 0 \]
We can rearrange this to isolate the constant on one side:
\[ x^2 + y^2 - 4x - 6y = -4 \]

### Step 2: Complete the square for \(x\) and \(y\)
Next, we complete the square for the \(x\) terms and the \(y\) terms.

For \(x^2 - 4x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

For \(y^2 - 6y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

Now, substituting these back into the equation:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 = -4 \]
This simplifies to:
\[ (x - 2)^2 + (y - 3)^2 - 13 = -4 \]
\[ (x - 2)^2 + (y - 3)^2 = 9 \]

### Step 3: Identify the center and radius
Now, we can identify the center and radius from the standard form of the circle \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

From \((x - 2)^2 + (y - 3)^2 = 9\):
- The center \((h, k) = (2, 3)\)
- The radius \(r = \sqrt{9} = 3\)

### Step 4: Calculate the diameter
The diameter \(D\) of a circle is given by:
\[ D = 2r \]
Substituting the radius we found:
\[ D = 2 \times 3 = 6 \]

### Final Answer
The length of the diameter of the circle is \(6\) units.

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