If (4, 1) be an end of a diameter of the circle `x^2 + y^2 - 2x + 6y-15=0`, find the coordinates of the other end of the diameter.

Answer

To find the coordinates of the other end of the diameter of the circle given that one end is (4, 1), we will follow these steps:

### Step 1: Identify the center of the circle
The equation of the circle is given as:
\[ x^2 + y^2 - 2x + 6y - 15 = 0 \]

We can rewrite this in the standard form of a circle, which is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

To do this, we will complete the square for the x and y terms.

**Completing the square for x:**
- The x terms are \( x^2 - 2x \).
- To complete the square, take half of -2 (which is -1), square it (getting 1), and add and subtract it:
\[ x^2 - 2x = (x - 1)^2 - 1 \]

**Completing the square for y:**
- The y terms are \( y^2 + 6y \).
- Take half of 6 (which is 3), square it (getting 9), and add and subtract it:
\[ y^2 + 6y = (y + 3)^2 - 9 \]

Now substitute these back into the equation:
\[ (x - 1)^2 - 1 + (y + 3)^2 - 9 - 15 = 0 \]
\[ (x - 1)^2 + (y + 3)^2 - 25 = 0 \]
\[ (x - 1)^2 + (y + 3)^2 = 25 \]

From this, we can see that the center of the circle \( C \) is at \( (h, k) = (1, -3) \).

### Step 2: Use the midpoint theorem
Let the other end of the diameter be \( P(a, b) \). The midpoint \( M \) of the diameter, which is also the center of the circle, can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Here, one end of the diameter is \( (4, 1) \) and the other end is \( (a, b) \). Therefore, we can set up the following equations based on the coordinates of the center:
1. For the x-coordinates:
\[ 1 = \frac{4 + a}{2} \]
2. For the y-coordinates:
\[ -3 = \frac{1 + b}{2} \]

### Step 3: Solve for a and b
**Solving the first equation:**
\[ 1 = \frac{4 + a}{2} \]
Multiply both sides by 2:
\[ 2 = 4 + a \]
Subtract 4 from both sides:
\[ a = 2 - 4 = -2 \]

**Solving the second equation:**
\[ -3 = \frac{1 + b}{2} \]
Multiply both sides by 2:
\[ -6 = 1 + b \]
Subtract 1 from both sides:
\[ b = -6 - 1 = -7 \]

### Conclusion
The coordinates of the other end of the diameter \( P \) are:
\[ P(-2, -7) \]

### Final Answer
The coordinates of the other end of the diameter are \( (-2, -7) \).

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