The equation ` 45 x^(2) + 45 y^(2) - 60 x + 36 y + 19 = 0` is equivalent to

Answer

To rewrite the equation \( 45x^2 + 45y^2 - 60x + 36y + 19 = 0 \) in a standard form of a circle, we will follow these steps:

### Step 1: Divide the entire equation by 45
To simplify the equation, we can divide every term by 45:

\[ x^2 + y^2 - \frac{60}{45}x + \frac{36}{45}y + \frac{19}{45} = 0 \]

### Step 2: Simplify the fractions
Now, we simplify the fractions:

\[ x^2 + y^2 - \frac{4}{3}x + \frac{4}{5}y + \frac{19}{45} = 0 \]

### Step 3: Rearrange the equation
Rearranging gives us:

\[ x^2 - \frac{4}{3}x + y^2 + \frac{4}{5}y = -\frac{19}{45} \]

### Step 4: Complete the square for \(x\)
To complete the square for \(x\), we take the coefficient of \(x\), which is \(-\frac{4}{3}\), halve it to get \(-\frac{2}{3}\), and then square it:

\[ \left(-\frac{2}{3}\right)^2 = \frac{4}{9} \]

Now, we add and subtract \(\frac{4}{9}\) inside the equation:

\[ \left(x - \frac{2}{3}\right)^2 - \frac{4}{9} + y^2 + \frac{4}{5}y = -\frac{19}{45} \]

### Step 5: Complete the square for \(y\)
Next, we complete the square for \(y\). The coefficient of \(y\) is \(\frac{4}{5}\). Halving it gives \(\frac{2}{5}\), and squaring it gives:

\[ \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]

We add and subtract \(\frac{4}{25}\):

\[ \left(x - \frac{2}{3}\right)^2 - \frac{4}{9} + \left(y + \frac{2}{5}\right)^2 - \frac{4}{25} = -\frac{19}{45} \]

### Step 6: Combine the constants
Now we combine the constants on the right side:

\[ \left(x - \frac{2}{3}\right)^2 + \left(y + \frac{2}{5}\right)^2 = -\frac{19}{45} + \frac{4}{9} + \frac{4}{25} \]

### Step 7: Find a common denominator and simplify
The common denominator for \(45\), \(9\), and \(25\) is \(225\). We convert each term:

\[ -\frac{19}{45} = -\frac{95}{225}, \quad \frac{4}{9} = \frac{100}{225}, \quad \frac{4}{25} = \frac{36}{225} \]

Now we can combine:

\[ -\frac{95}{225} + \frac{100}{225} + \frac{36}{225} = \frac{41}{225} \]

### Final Equation
Thus, the equation of the circle is:

\[ \left(x - \frac{2}{3}\right)^2 + \left(y + \frac{2}{5}\right)^2 = \frac{41}{225} \]