If the coordinate of one end of a diameter of the circle `x^2+y^2+4x−8y+5=0` is (2,1), the coordinates of the other end is a) (-6,--7) b) (6,7) c) (-6,7) d) (7,-6)

Answer

To find the coordinates of the other end of the diameter of the circle given one end at (2, 1), we will follow these steps:

1. **Identify the center of the circle**: The equation of the circle is given as \(x^2 + y^2 + 4x - 8y + 5 = 0\). We need to rewrite this in standard form to find the center.

2. **Complete the square**:
- For the \(x\) terms: \(x^2 + 4x\) can be rewritten as \((x + 2)^2 - 4\).
- For the \(y\) terms: \(y^2 - 8y\) can be rewritten as \((y - 4)^2 - 16\).

3. **Rewrite the equation**:
\[ (x + 2)^2 - 4 + (y - 4)^2 - 16 + 5 = 0 \]
Simplifying this gives:
\[ (x + 2)^2 + (y - 4)^2 - 15 = 0 \implies (x + 2)^2 + (y - 4)^2 = 15 \]
This shows that the center of the circle is at \((-2, 4)\).

4. **Use the midpoint formula**: The center of the circle is the midpoint of the diameter. If one end of the diameter is \((2, 1)\) and the other end is \((h, k)\), then:
\[ \left(\frac{2 + h}{2}, \frac{1 + k}{2}\right) = (-2, 4) \]

5. **Set up equations**:
- From the x-coordinates:
\[ \frac{2 + h}{2} = -2 \implies 2 + h = -4 \implies h = -6 \]
- From the y-coordinates:
\[ \frac{1 + k}{2} = 4 \implies 1 + k = 8 \implies k = 7 \]

6. **Conclusion**: The coordinates of the other end of the diameter are \((-6, 7)\).

Thus, the answer is \((-6, 7)\), which corresponds to option (c).