Find the pair of tangents from the origin
to the circle `x^(2) + y^(2) + 2gx + 2fy + c = 0`
and hence deduce a condition for these
tangents to be perpendicular.

Answer

To find the pair of tangents from the origin to the circle given by the equation \(x^2 + y^2 + 2gx + 2fy + c = 0\) and to deduce the condition for these tangents to be perpendicular, we can follow these steps:

### Step 1: Identify the equation of the circle
The equation of the circle is given as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

### Step 2: Use the formula for the pair of tangents from a point to a circle
The formula for the pair of tangents from a point \((x_1, y_1)\) to the circle is:
\[ SS_1 = T^2 \]
where \(S\) is the equation of the circle, \(S_1\) is the equation of the circle when the point \((x_1, y_1)\) is substituted, and \(T^2\) is given by:
\[ T^2 = xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c \]

### Step 3: Substitute the origin into the equations
For the origin \((0, 0)\):
- \(S_1 = 0^2 + 0^2 + 2g(0) + 2f(0) + c = c\)
- \(T^2 = x(0) + y(0) + g(x + 0) + f(y + 0) + c = gx + fy + c\)

Thus, we have:
\[ S \cdot S_1 = T^2 \implies (x^2 + y^2 + 2gx + 2fy + c) \cdot c = (gx + fy + c)^2 \]

### Step 4: Expand both sides
Expanding the left-hand side:
\[ c(x^2 + y^2 + 2gx + 2fy + c) = cx^2 + cy^2 + 2cgx + 2cfy + c^2 \]

Expanding the right-hand side:
\[ (gx + fy + c)^2 = g^2x^2 + f^2y^2 + c^2 + 2gxc + 2fyc \]

### Step 5: Set the equations equal
Setting both expansions equal:
\[ cx^2 + cy^2 + 2cgx + 2cfy + c^2 = g^2x^2 + f^2y^2 + c^2 + 2gxc + 2fyc \]

### Step 6: Cancel terms
Cancelling \(c^2\) and \(2gxc\) and \(2fyc\) from both sides gives:
\[ cx^2 + cy^2 = g^2x^2 + f^2y^2 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ (c - g^2)x^2 + (c - f^2)y^2 = 0 \]

### Step 8: Condition for tangents to be perpendicular
The above equation represents a pair of straight lines. For these lines to be perpendicular, the sum of the coefficients of \(x^2\) and \(y^2\) must be zero:
\[ (c - g^2) + (c - f^2) = 0 \]
This simplifies to:
\[ 2c = g^2 + f^2 \]

### Conclusion
Thus, the condition for the tangents from the origin to the circle to be perpendicular is:
\[ 2c = g^2 + f^2 \]

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