Find the length of the chord intercepted
by the circle `x^(2) + y^(2) - x + 3y- 22 = 0` on
the line `y = x - 3 `

Answer

To find the length of the chord intercepted by the circle given by the equation \( x^2 + y^2 - x + 3y - 22 = 0 \) on the line \( y = x - 3 \), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - x + 3y - 22 = 0 \]
We can rearrange this to:
\[ x^2 - x + y^2 + 3y - 22 = 0 \]

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

For \( x \):
\[ x^2 - x = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} \]

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

Substituting these back into the equation gives:
\[ \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} + \left(y + \frac{3}{2}\right)^2 - \frac{9}{4} - 22 = 0 \]
Combining the constants:
\[ \left(x - \frac{1}{2}\right)^2 + \left(y + \frac{3}{2}\right)^2 = 22 + \frac{1}{4} + \frac{9}{4} = 22 + \frac{10}{4} = 22 + 2.5 = 24.5 \]

Thus, the center of the circle is \( \left(\frac{1}{2}, -\frac{3}{2}\right) \) and the radius is \( \sqrt{24.5} \).

### Step 3: Substitute the Line Equation into the Circle Equation
Now we substitute the line equation \( y = x - 3 \) into the circle equation to find the points of intersection.

Substituting \( y \):
\[ x^2 + (x - 3)^2 - x + 3(x - 3) - 22 = 0 \]
Expanding this:
\[ x^2 + (x^2 - 6x + 9) - x + 3x - 9 - 22 = 0 \]
Combining like terms:
\[ 2x^2 - 4x - 22 = 0 \]
Dividing by 2:
\[ x^2 - 2x - 11 = 0 \]

### Step 4: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -2, c = -11 \):
\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-11)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 44}}{2} = \frac{2 \pm \sqrt{48}}{2} = \frac{2 \pm 4\sqrt{3}}{2} = 1 \pm 2\sqrt{3} \]

### Step 5: Find Corresponding y-values
Now we find the corresponding \( y \)-values using \( y = x - 3 \):
1. For \( x_1 = 1 + 2\sqrt{3} \):
\[ y_1 = (1 + 2\sqrt{3}) - 3 = -2 + 2\sqrt{3} \]
2. For \( x_2 = 1 - 2\sqrt{3} \):
\[ y_2 = (1 - 2\sqrt{3}) - 3 = -2 - 2\sqrt{3} \]

### Step 6: Calculate the Length of the Chord
Now we have the points of intersection:
\[ (x_1, y_1) = (1 + 2\sqrt{3}, -2 + 2\sqrt{3}), \quad (x_2, y_2) = (1 - 2\sqrt{3}, -2 - 2\sqrt{3}) \]

Using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating:
\[ x_2 - x_1 = (1 - 2\sqrt{3}) - (1 + 2\sqrt{3}) = -4\sqrt{3} \]
\[ y_2 - y_1 = (-2 - 2\sqrt{3}) - (-2 + 2\sqrt{3}) = -4\sqrt{3} \]
Thus:
\[ \text{Length} = \sqrt{(-4\sqrt{3})^2 + (-4\sqrt{3})^2} = \sqrt{48 + 48} = \sqrt{96} = 4\sqrt{6} \]

### Final Answer
The length of the chord is \( 4\sqrt{6} \) units.

---