To solve the problem, we will analyze both statements step by step.
### Statement I:
**Claim:** The circle with the points of intersection of the line \(3x + 4y = 12\) with axes as extremities of a diameter is \(x^2 + y^2 - 4x - 3y = 0\).
1. **Find the intercepts of the line \(3x + 4y = 12\):**
- To find the x-intercept, set \(y = 0\):
\[
3x + 4(0) = 12 \implies x = 4 \implies \text{Point: } (4, 0)
\]
- To find the y-intercept, set \(x = 0\):
\[
3(0) + 4y = 12 \implies y = 3 \implies \text{Point: } (0, 3)
\]
2. **Identify the points as extremities of the diameter:**
- The points of intersection are \(A(4, 0)\) and \(B(0, 3)\).
3. **Find the center of the circle:**
- The center \(C\) of the circle is the midpoint of \(A\) and \(B\):
\[
C\left(\frac{4 + 0}{2}, \frac{0 + 3}{2}\right) = C(2, 1.5)
\]
4. **Calculate the radius:**
- The radius \(r\) is half the distance between \(A\) and \(B\):
\[
r = \frac{1}{2} \sqrt{(4 - 0)^2 + (0 - 3)^2} = \frac{1}{2} \sqrt{16 + 9} = \frac{1}{2} \sqrt{25} = \frac{5}{2}
\]
5. **Write the equation of the circle:**
- The equation of the circle in standard form is:
\[
(x - 2)^2 + (y - 1.5)^2 = \left(\frac{5}{2}\right)^2
\]
- Expanding this gives:
\[
(x - 2)^2 + (y - 1.5)^2 = \frac{25}{4}
\]
- Converting to general form:
\[
x^2 - 4x + 4 + y^2 - 3y + \frac{9}{4} - \frac{25}{4} = 0
\]
- Simplifying:
\[
x^2 + y^2 - 4x - 3y = 0
\]
6. **Conclusion for Statement I:**
- Since the derived equation matches the given equation, **Statement I is true**.
### Statement II:
**Claim:** The circle passing through \((0, 0)\) and making intercepts 8 and 6 on the x and y axes has its center at \((-4, 2)\).
1. **Identify the intercepts:**
- The x-intercept is \(8\) (point \((8, 0)\)) and the y-intercept is \(6\) (point \((0, 6)\)).
2. **Find the center of the circle:**
- The center of the circle is the midpoint of the intercepts:
\[
C\left(\frac{8 + 0}{2}, \frac{0 + 6}{2}\right) = C(4, 3)
\]
3. **Check the given center:**
- The given center is \((-4, 2)\), which does not match the calculated center \((4, 3)\).
4. **Conclusion for Statement II:**
- Since the calculated center does not match the given center, **Statement II is false**.
### Final Conclusion:
- **Statement I is true.**
- **Statement II is false.**
