The condition that the pair of tangents drawn from origin to circle
`x^(2)+y^(2)+2gx+2fy+c=0` may be at right angles is

Options

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

Answer

To find the condition that the pair of tangents drawn from the origin to the circle given by the equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \) are at right angles, we can follow these steps:

### Step 1: Understand the Circle Equation
The given equation of the circle is in the general form:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this, we can identify the center of the circle as \( (-g, -f) \) and the radius \( r \) as:
\[ r = \sqrt{g^2 + f^2 - c} \]

### Step 2: Condition for Right Angled Tangents
For the tangents from a point (in this case, the origin) to be at right angles, the following condition must hold:
\[ g^2 + f^2 = 2c \]
This is derived from the property of the director circle, which is the locus of the intersection of the pair of perpendicular tangents.

### Step 3: Deriving the Condition
To derive this condition, we can consider the equation of the director circle, which is given by:
\[ (x - g)^2 + (y - f)^2 = 2r^2 \]
Substituting \( r^2 \) from our earlier calculation, we have:
\[ (x - g)^2 + (y - f)^2 = 2(g^2 + f^2 - c) \]

### Step 4: Substitute the Origin
Now, substituting the origin \( (0, 0) \) into the director circle equation:
\[ (0 - g)^2 + (0 - f)^2 = 2(g^2 + f^2 - c) \]
This simplifies to:
\[ g^2 + f^2 = 2(g^2 + f^2 - c) \]

### Step 5: Simplifying the Equation
Expanding the right side:
\[ g^2 + f^2 = 2g^2 + 2f^2 - 2c \]
Rearranging gives:
\[ g^2 + f^2 - 2g^2 - 2f^2 + 2c = 0 \]
This simplifies to:
\[ -g^2 - f^2 + 2c = 0 \]
or:
\[ g^2 + f^2 = 2c \]

### Conclusion
Thus, the condition that the pair of tangents drawn from the origin to the circle are at right angles is:
\[ g^2 + f^2 = 2c \]