If one end of a diameter of the circle `x^(2) + y^(2) - 4x - 6y + 11 = 0 ` is (3, 4), then the coordinates of the other end of the diameter are

Options

A. (a) (2,1)
B. (b) (-2,1)
C. (c) (1,2)
D. (d) (-1,-2)

Answer

To find the coordinates of the other end of the diameter of the circle given by the equation \(x^2 + y^2 - 4x - 6y + 11 = 0\) with one end of the diameter at the point \(A(3, 4)\), we can follow these steps:

### Step 1: Rewrite the equation of the circle in standard form
The general form of the equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation, we have:
\[ x^2 + y^2 - 4x - 6y + 11 = 0 \]
We can identify \(2g = -4\), \(2f = -6\), and \(c = 11\).

### Step 2: Find the center of the circle
From the coefficients, we can find \(g\) and \(f\):
\[ g = -2 \quad \text{and} \quad f = -3 \]
The center \(C\) of the circle is given by:
\[ C = (-g, -f) = (2, 3) \]

### Step 3: Use the midpoint formula
Let the coordinates of the other end of the diameter be \(B(x, y)\). The center \(C\) is the midpoint of the diameter \(AB\). Therefore, we can use the midpoint formula:
\[ C = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the known values:
\[ (2, 3) = \left(\frac{3 + x}{2}, \frac{4 + y}{2}\right) \]

### Step 4: Set up equations from the midpoint
From the x-coordinates:
\[ 2 = \frac{3 + x}{2} \]
Multiplying both sides by 2:
\[ 4 = 3 + x \implies x = 1 \]

From the y-coordinates:
\[ 3 = \frac{4 + y}{2} \]
Multiplying both sides by 2:
\[ 6 = 4 + y \implies y = 2 \]

### Step 5: Conclusion
Thus, the coordinates of the other end of the diameter \(B\) are:
\[ B(1, 2) \]

### Final Answer
The coordinates of the other end of the diameter are \( (1, 2) \).

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