A variable chord of circle `x^(2)+y^(2)+2gx+2fy+c=0` passes through the point `P(x_(1),y_(1))`. Find the locus of the midpoint of the chord.

Explanation

To find the locus of the midpoint of a variable chord of the circle given by the equation \(x^2 + y^2 + 2gx + 2fy + c = 0\) that passes through the point \(P(x_1, y_1)\), we can follow these steps:

### Step 1: Define the midpoint of the chord
Let \(H\) and \(K\) be the coordinates of the midpoint of the chord. Thus, we can denote the midpoint as \((H, K)\).

### Step 2: Write the equation of the chord
The equation of the chord can be expressed in terms of the midpoint \((H, K)\) and the circle's equation. The general form of the chord passing through a point \(P(x_1, y_1)\) can be written as:
\[
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