No portion of the circle `x ^(2) + y^(2) - 16x + 18y + 1=0` lies in the

Options

A. first quadrnt
B. second quadrant
C. third quadrant
D. fourth quadrant

Answer

To solve the problem, we need to analyze the given equation of the circle and determine its center and radius. The equation provided is:

\[ x^2 + y^2 - 16x + 18y + 1 = 0 \]

### Step 1: Rewrite the equation in standard form

To find the center and radius of the circle, we need to rewrite the equation in standard form by completing the square for both x and y.

1. Group the x and y terms:
\[ (x^2 - 16x) + (y^2 + 18y) + 1 = 0 \]

2. Complete the square for the x terms:
- Take half of -16, square it: \((-8)^2 = 64\)
- Add and subtract 64:
\[ (x^2 - 16x + 64 - 64) = (x - 8)^2 - 64 \]

3. Complete the square for the y terms:
- Take half of 18, square it: \((9)^2 = 81\)
- Add and subtract 81:
\[ (y^2 + 18y + 81 - 81) = (y + 9)^2 - 81 \]

4. Substitute back into the equation:
\[ (x - 8)^2 - 64 + (y + 9)^2 - 81 + 1 = 0 \]

5. Simplify:
\[ (x - 8)^2 + (y + 9)^2 - 144 = 0 \]
\[ (x - 8)^2 + (y + 9)^2 = 144 \]

### Step 2: Identify the center and radius

From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center \((h, k) = (8, -9)\)
- Radius \(r = \sqrt{144} = 12\)

### Step 3: Determine the position of the circle

1. The center of the circle is at \((8, -9)\), which is located in the fourth quadrant (x positive, y negative).
2. The radius is 12, which means the circle extends 12 units from the center in all directions.

### Step 4: Analyze the quadrants

To determine which quadrants the circle occupies:
- The center is at (8, -9).
- The circle extends 12 units up and down, and left and right from the center.

**Vertical extent:**
- From \(y = -9 - 12 = -21\) to \(y = -9 + 12 = 3\) (covers the fourth and third quadrants).

**Horizontal extent:**
- From \(x = 8 - 12 = -4\) to \(x = 8 + 12 = 20\) (covers the first and second quadrants).

### Step 5: Conclusion

The circle occupies parts of the first, second, third, and fourth quadrants. However, it does not extend into the second quadrant because the lowest point of the circle (y = -21) is below the x-axis and does not reach into the positive y-values of the second quadrant.

Thus, the portion of the circle that does not lie in the second quadrant is confirmed.

### Final Answer

No portion of the circle lies in the **second quadrant**.

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