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To solve this problem, we need to find the value of \( q \) for a circle centered at \( (3, 1) \) with points on the circle at \( (0, 5) \) and \( (q, 0) \).

Let's solve it step-by-step:

1. **Calculate the Radius of the Circle:**
   The radius \( r \) of a circle is the distance from the center to any point on the circumference. We can use the point \( (0, 5) \) for this purpose.

   The distance formula between \( (3, 1) \) and \( (0, 5) \) is:
   \[
   r = \sqrt{(3 - 0)^2 + (1 - 5)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
   \]

   So, the radius \( r \) is 5.

2. **Set Up the Circle Equation:**
   With the center at \( (3, 1) \) and radius 5, the equation of the circle is:
   \[
   (x - 3)^2 + (y - 1)^2 = 25
   \]

3. **Substitute \( y = 0 \) to Find \( q \):**
   We know that \( (q, 0) \) is a point on the circle, so substitute \( y = 0 \) into the circle equation to find \( q \):
   \[
   (q - 3)^2 + (0 - 1)^2 = 25
   \]
   Simplifying, we get:
   \[
   (q - 3)^2 + 1 = 25
   \]
   \[
   (q - 3)^2 = 24
   \]
   \[
   q - 3 = \pm \sqrt{24}
   \]
   \[
   q = 3 \pm \sqrt{24}
   \]

4. **Simplify \( \sqrt{24} \):**
   Since \( \sqrt{24} = 2\sqrt{6} \), we have:
   \[
   q = 3 \pm 2\sqrt{6}
   \]

5. **Choose the Correct Form:**
   To express the answer in terms of a positive surd, the positive form \( 3 + 2\sqrt{6} \) should be used.

### Final Answer:
\[
q = 3 + 2\sqrt{6}
\]

Would you like further details on any step, or do you have other questions?

### Related Questions:
1. How do we determine the equation of a circle given a different center and radius?
2. What is the distance formula and how is it derived?
3. How do you simplify square roots, like turning \( \sqrt{24} \) into \( 2\sqrt{6} \)?
4. How would the answer change if the center were at a different point?
5. What are other ways to determine if a point lies on a given circle?

#### Tip:
To simplify square roots like \( \sqrt{24} \), break it down into prime factors to find pairs, which helps rewrite it as \( 2\sqrt{6} \).

The central point of a circle in xy-plane is (3,1). There is another point which is located at the edge of the circle and its coordinate is (0,5). Furthermore, there is another point which is at the edge of the circle and its coordinates is (q, 0). Evaluate the value of q in a form of positive surd.

Learn how to evaluate the value of q for a circle centered at (3,1) with points on the edge at (0,5) and (q,0). The solution involves applying the circle equation and simplifying surds. Suitable for high school students in Grades 9-12.

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