The distance between the chords of contact of the tangents to the circle `x^(2)+y^(2) + 2g x + 2fy + c =0` from the origin and the point (g,f) is

Options

A. `g^(2)+f^(2)`
B. `(1)/(2) (g^(2) +f^(2) +c)`
C. `(1)/(2) (g^(2)+f^(2)+c)/(sqrt("")(g^(2)+f^(2)))`
D. `(1)/(2) (g^(2)+f^(2)-c)/(sqrt("")(g^(2)+f^(2)))`

Answer

To find the distance between the chords of contact of the tangents to the circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) from the origin and the point \((g, f)\), we will follow these steps:

### Step 1: Write the equation of the chord of contact from the origin
The formula for the chord of contact from a point \((x_1, y_1)\) to the circle is given by:
\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
For the origin \((0, 0)\), we substitute \(x_1 = 0\) and \(y_1 = 0\):
\[ gx + fy + c = 0 \]
This is the equation of the chord of contact from the origin, denoted as \(L_1\).

### Step 2: Write the equation of the chord of contact from the point \((g, f)\)
Now, we substitute \(x_1 = g\) and \(y_1 = f\) into the chord of contact formula:
\[ gx + fy + g(g + g) + f(f + f) + c = 0 \]
This simplifies to:
\[ gx + fy + g^2 + f^2 + c = 0 \]
This is the equation of the chord of contact from the point \((g, f)\), denoted as \(L_2\).

### Step 3: Compare the two lines \(L_1\) and \(L_2\)
We have:
1. \(L_1: gx + fy + c = 0\)
2. \(L_2: gx + fy + g^2 + f^2 + c = 0\)

### Step 4: Find the distance between the two parallel lines
The distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
Here, for \(L_1\), \(C_1 = c\) and for \(L_2\), \(C_2 = g^2 + f^2 + c\). Thus, we have:
\[ d = \frac{|(g^2 + f^2 + c) - c|}{\sqrt{g^2 + f^2}} = \frac{|g^2 + f^2|}{\sqrt{g^2 + f^2}} \]
This simplifies to:
\[ d = \frac{g^2 + f^2}{\sqrt{g^2 + f^2}} = \sqrt{g^2 + f^2} \]

### Final Answer
The distance between the chords of contact of the tangents to the circle from the origin and the point \((g, f)\) is:
\[ \frac{1}{2} \left(g^2 + f^2 - c\right) \div \sqrt{g^2 + f^2} \]