The condition so that the line
`(x+g) cos theta +(y+f)sin theta =k`
is a tangent to `x^(2)+y^(2) +2gx +2fy +c=0` is

Options

A. `g^(2)+f^(2) = c+k^(2)`
B. `g^(2)+f^(2) =c^(2)+k`
C. `g^(2)+f^(2)=c^(2)+k^(2)`
D. `g^(2)+f^(2)=c+k`

Answer

To find the condition for the line \((x + g) \cos \theta + (y + f) \sin \theta = k\) to be a tangent to the circle defined by the equation \(x^2 + y^2 + 2gx + 2fy + c = 0\), we can follow these steps:

### Step 1: Identify the Circle's Center and Radius
The general equation of a circle is given as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this equation, we can identify:
- Center of the circle: \((-g, -f)\)
- Radius of the circle:
\[ r = \sqrt{g^2 + f^2 - c} \]

### Step 2: Write the Equation of the Tangent Line
The equation of the line is given as:
\[ (x + g) \cos \theta + (y + f) \sin \theta = k \]
We can rewrite this in the standard form:
\[ (x \cos \theta + y \sin \theta) + (g \cos \theta + f \sin \theta - k) = 0 \]
This can be expressed as:
\[ Ax + By + C = 0 \]
where:
- \(A = \cos \theta\)
- \(B = \sin \theta\)
- \(C = g \cos \theta + f \sin \theta - k\)

### Step 3: Calculate the Perpendicular Distance from the Center to the Line
The distance \(d\) from the center of the circle \((-g, -f)\) to the line \(Ax + By + C = 0\) is given by the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting the center coordinates \((-g, -f)\):
\[ d = \frac{|\cos \theta (-g) + \sin \theta (-f) + (g \cos \theta + f \sin \theta - k)|}{\sqrt{\cos^2 \theta + \sin^2 \theta}} \]
Since \(\cos^2 \theta + \sin^2 \theta = 1\), we simplify:
\[ d = | -k | = k \]

### Step 4: Set the Distance Equal to the Radius
For the line to be a tangent to the circle, the distance from the center to the line must equal the radius of the circle:
\[ k = \sqrt{g^2 + f^2 - c} \]

### Step 5: Square Both Sides
Squaring both sides gives:
\[ k^2 = g^2 + f^2 - c \]

### Step 6: Rearranging the Equation
Rearranging the equation leads to:
\[ k^2 + c = g^2 + f^2 \]

### Conclusion
Thus, the condition for the line to be a tangent to the circle is:
\[ g^2 + f^2 = k^2 + c \]