If the coordinates of one end of a diameter of the circles `x^(2)+y^(2)+4x-8y+5=0` is (2,1),, the coordinates of the other end is

Options

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 by the equation \(x^2 + y^2 + 4x - 8y + 5 = 0\) when one end is (2, 1), we can follow these steps:

### Step 1: Rewrite the Circle Equation
The given equation of the circle is:
\[ x^2 + y^2 + 4x - 8y + 5 = 0 \]
We can rewrite it in the standard form by completing the square.

### Step 2: Complete the Square
1. For \(x\):
\[ x^2 + 4x = (x + 2)^2 - 4 \]
2. For \(y\):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

Substituting these back into the equation:
\[ (x + 2)^2 - 4 + (y - 4)^2 - 16 + 5 = 0 \]
This simplifies to:
\[ (x + 2)^2 + (y - 4)^2 - 15 = 0 \]
So, we can rewrite it as:
\[ (x + 2)^2 + (y - 4)^2 = 15 \]
This shows that the center of the circle is at \((-2, 4)\) and the radius is \(\sqrt{15}\).

### Step 3: Find the Midpoint
The midpoint of the diameter is the center of the circle. The center is given by the coordinates:
\[ \left(-2, 4\right) \]
Let the coordinates of the other end of the diameter be \((h, k)\). The midpoint formula states:
\[ \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(-2, 4\right) \]
where \((x_1, y_1) = (2, 1)\) and \((x_2, y_2) = (h, k)\).

### Step 4: Set Up the Equations
From the midpoint formula, we can set up the following equations:
1. For the x-coordinates:
\[ \frac{2 + h}{2} = -2 \]
Multiplying both sides by 2:
\[ 2 + h = -4 \implies h = -4 - 2 = -6 \]

2. For the y-coordinates:
\[ \frac{1 + k}{2} = 4 \]
Multiplying both sides by 2:
\[ 1 + k = 8 \implies k = 8 - 1 = 7 \]

### Step 5: Conclusion
Thus, the coordinates of the other end of the diameter are:
\[ (h, k) = (-6, 7) \]

### Final Answer
The coordinates of the other end of the diameter are \((-6, 7)\).

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