If the length of the tangent from (f,g) to the circle `x^(2)+y^(2)=6` is twice the length of the tangent from the same point to the circle `x^(2)+y^(2)+3x+3y=0`, then

Options

A. `f^(2)+g^(2)+4f+4g-2=0`
B. `f^(2)+g^(2)+4f+4g+2=0`
C. `f^(2)-g^(2)+4f=0`
D. `f^(2)+g^(2)+4f=0`

Answer

To solve the problem, we need to find the relationship between the lengths of tangents from the point (f, g) to two circles given by the equations \(x^2 + y^2 = 6\) and \(x^2 + y^2 + 3x + 3y = 0\).

### Step 1: Identify the equations of the circles
The first circle is given by:
\[ x^2 + y^2 = 6 \]
This can be rewritten as:
\[ x^2 + y^2 - 6 = 0 \]
The second circle is given by:
\[ x^2 + y^2 + 3x + 3y = 0 \]
This can be rewritten as:
\[ x^2 + y^2 + 3x + 3y = 0 \]

### Step 2: Calculate the length of the tangent from point (f, g) to the first circle
The formula for the length of the tangent from a point \((f, g)\) to a circle defined by \(S_1 = 0\) is given by:
\[ L_1 = \sqrt{S_1} = \sqrt{f^2 + g^2 - 6} \]

### Step 3: Calculate the length of the tangent from point (f, g) to the second circle
For the second circle, we can express it in the form \(S_2 = 0\):
\[ S_2 = f^2 + g^2 + 3f + 3g \]
Thus, the length of the tangent from point \((f, g)\) to the second circle is:
\[ L_2 = \sqrt{S_2} = \sqrt{f^2 + g^2 + 3f + 3g} \]

### Step 4: Set up the relationship between the lengths of tangents
According to the problem, the length of the tangent from (f, g) to the first circle is twice the length of the tangent to the second circle:
\[ L_1 = 2L_2 \]

### Step 5: Substitute the lengths of tangents into the equation
Substituting the expressions for \(L_1\) and \(L_2\):
\[ \sqrt{f^2 + g^2 - 6} = 2\sqrt{f^2 + g^2 + 3f + 3g} \]

### Step 6: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ f^2 + g^2 - 6 = 4(f^2 + g^2 + 3f + 3g) \]

### Step 7: Expand and simplify the equation
Expanding the right side:
\[ f^2 + g^2 - 6 = 4f^2 + 4g^2 + 12f + 12g \]
Rearranging gives:
\[ f^2 + g^2 - 4f^2 - 4g^2 - 12f - 12g - 6 = 0 \]
This simplifies to:
\[ -3f^2 - 3g^2 - 12f - 12g - 6 = 0 \]

### Step 8: Factor out the common terms
Dividing the entire equation by -3:
\[ f^2 + g^2 + 4f + 4g + 2 = 0 \]

### Final Step: Write the final relation
Thus, the relation that holds true is:
\[ f^2 + g^2 + 4f + 4g + 2 = 0 \]