Suppose the equation of circle is `x^2 + y^2 - 8x - 6y + 24 = 0` and let `(p, q)` is any point on the circle, Then the no. of possible integral values of `|p+q|` is

Answer

To solve the problem, we need to analyze the given equation of the circle and find the integral values of \( |p + q| \) for points \( (p, q) \) on the circle.

### Step 1: Rewrite the Circle's Equation
The given equation of the circle is:
\[ x^2 + y^2 - 8x - 6y + 24 = 0 \]
We can rearrange this equation by completing the square for both \( x \) and \( y \).

### Step 2: Complete the Square
For \( x \):
\[ x^2 - 8x = (x - 4)^2 - 16 \]
For \( y \):
\[ y^2 - 6y = (y - 3)^2 - 9 \]
Substituting these into the equation gives:
\[ (x - 4)^2 - 16 + (y - 3)^2 - 9 + 24 = 0 \]
This simplifies to:
\[ (x - 4)^2 + (y - 3)^2 - 1 = 0 \]
Thus, we have:
\[ (x - 4)^2 + (y - 3)^2 = 1 \]
This represents a circle with center \( (4, 3) \) and radius \( 1 \).

### Step 3: Identify Points on the Circle
The points on the circle can be expressed as:
\[ p = 4 + \cos(\theta), \quad q = 3 + \sin(\theta) \]
for \( \theta \) in \( [0, 2\pi) \).

### Step 4: Calculate \( |p + q| \)
We need to find:
\[ |p + q| = |(4 + \cos(\theta)) + (3 + \sin(\theta))| = |7 + \cos(\theta) + \sin(\theta)| \]
Let \( z = \cos(\theta) + \sin(\theta) \). The maximum and minimum values of \( z \) can be found using the identity:
\[ \cos(\theta) + \sin(\theta) = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
The maximum value of \( z \) is \( \sqrt{2} \) and the minimum value is \( -\sqrt{2} \).

### Step 5: Find the Range of \( |7 + z| \)
Thus, the range of \( z \) is:
\[ -\sqrt{2} \leq z \leq \sqrt{2} \]
This leads to:
\[ 7 - \sqrt{2} \leq 7 + z \leq 7 + \sqrt{2} \]
Calculating the approximate values:
- \( 7 - \sqrt{2} \approx 7 - 1.414 \approx 5.586 \)
- \( 7 + \sqrt{2} \approx 7 + 1.414 \approx 8.414 \)

### Step 6: Determine Integral Values of \( |p + q| \)
Now, we need to find the integral values of \( |7 + z| \) within the range \( [5.586, 8.414] \). The possible integral values are:
- \( 6 \)
- \( 7 \)
- \( 8 \)

### Conclusion
Thus, the number of possible integral values of \( |p + q| \) is:
\[ \boxed{3} \]