The lines joining the origin to the point of intersection of `x^2+y^2+2gx+c=0 and x^2+y^2+2fy-c=0` are at right angles if

Answer

To solve the problem, we need to find the condition under which the lines joining the origin to the points of intersection of the two circles are at right angles. The equations of the circles are given as:

1. \( S_1: x^2 + y^2 + 2gx + c = 0 \)
2. \( S_2: x^2 + y^2 + 2fy - c = 0 \)

### Step-by-Step Solution:

**Step 1: Subtract the two circle equations.**

We start by subtracting the second equation from the first:

\[ S_1 - S_2 = (x^2 + y^2 + 2gx + c) - (x^2 + y^2 + 2fy - c) = 0 \]

This simplifies to:

\[ 2gx + 2c - 2fy = 0 \]

**Step 2: Rearranging the equation.**

Rearranging gives us:

\[ 2gx - 2fy + 2c = 0 \quad \Rightarrow \quad gx - fy + c = 0 \]

**Step 3: Expressing one variable in terms of the other.**

From the equation \( gx - fy + c = 0 \), we can express \( y \) in terms of \( x \):

\[ fy = gx + c \quad \Rightarrow \quad y = \frac{gx + c}{f} \]

**Step 4: Finding the intersection points.**

To find the points of intersection, we substitute \( y \) back into one of the original circle equations. Using \( S_1 \):

\[ x^2 + \left(\frac{gx + c}{f}\right)^2 + 2gx + c = 0 \]

Expanding this gives:

\[ x^2 + \frac{(gx + c)^2}{f^2} + 2gx + c = 0 \]

**Step 5: Collecting terms.**

This leads to a quadratic equation in \( x \):

\[ x^2 + 2gx + c + \frac{g^2x^2 + 2gcx + c^2}{f^2} = 0 \]

Combining like terms:

\[ \left(1 + \frac{g^2}{f^2}\right)x^2 + \left(2g + \frac{2gc}{f^2}\right)x + \left(c + \frac{c^2}{f^2}\right) = 0 \]

**Step 6: Condition for perpendicularity.**

For the lines joining the origin to the points of intersection to be at right angles, the coefficients of \( x^2 \) and \( y^2 \) in the resulting quadratic must satisfy the condition:

\[ \text{Coefficient of } x^2 + \text{Coefficient of } y^2 = 0 \]

This leads us to:

\[ 1 - \frac{2g^2}{c} + 1 + \frac{f^2}{c} = 0 \]

**Step 7: Simplifying the condition.**

Simplifying gives:

\[ 2 - \frac{g^2 - f^2}{c} = 0 \quad \Rightarrow \quad g^2 - f^2 = 2c \]

Thus, the condition for the lines joining the origin to the points of intersection of the circles to be at right angles is:

\[ g^2 - f^2 = 2c \]