The chord of contact of (1,2) with respect to the circle `x^(2)+y^(2)-4x-6y+2=0` is

Options

A. `x+y-6=0`
B. `x+2y-2=0`
C. `2x+y+6=0`
D. not existing

Answer

To find the chord of contact of the point (1, 2) with respect to the circle given by the equation \(x^2 + y^2 - 4x - 6y + 2 = 0\), we will follow these steps:

### Step 1: Write the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 6y + 2 = 0 \]

### Step 2: Identify the coefficients
We can compare the given circle equation with the general form of a circle:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify:
- \(2g = -4 \Rightarrow g = -2\)
- \(2f = -6 \Rightarrow f = -3\)
- \(c = 2\)

### Step 3: Use the formula for the chord of contact
The formula for the chord of contact from a point \((x_1, y_1)\) to the circle is given by:
\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
Here, \((x_1, y_1) = (1, 2)\).

### Step 4: Substitute the values into the chord of contact formula
Substituting \(x_1 = 1\), \(y_1 = 2\), \(g = -2\), \(f = -3\), and \(c = 2\) into the formula:
\[ x(1) + y(2) + (-2)(x + 1) + (-3)(y + 2) + 2 = 0 \]

### Step 5: Simplify the equation
Now, simplifying the equation:
\[ x + 2y - 2(x + 1) - 3(y + 2) + 2 = 0 \]
Expanding this:
\[ x + 2y - 2x - 2 - 3y - 6 + 2 = 0 \]
Combining like terms:
\[ -x - y - 6 = 0 \]
Rearranging gives:
\[ x + y + 6 = 0 \]

### Final Result
Thus, the equation of the chord of contact is:
\[ x + y + 6 = 0 \]