Shwo that the difference of the squares of the tangents to two coplanar circles from any point P in the plane of the circles varies as the perpendicular from P on their radical axis. Also, prove that the locus of a point such that the difference of the squares of the tangents from it to two given circles is constant is a line parallel to their radical axis.

Answer

To solve the given problem, we will break it down into two parts as described in the question.

### Part 1: Show that the difference of the squares of the tangents to two coplanar circles from any point P in the plane of the circles varies as the perpendicular from P on their radical axis.

1. **Let the two circles be defined as follows:**
- Circle 1: \( S_1: x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0 \)
- Circle 2: \( S_2: x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0 \)

2. **Let P be a point in the plane of the circles with coordinates (x, y).** The lengths of the tangents from point P to the circles can be expressed as:
- \( PA = \sqrt{S_1} = \sqrt{x^2 + y^2 + 2g_1x + 2f_1y + c_1} \)
- \( PB = \sqrt{S_2} = \sqrt{x^2 + y^2 + 2g_2x + 2f_2y + c_2} \)

3. **The difference of the squares of the tangents from point P to the circles is given by:**
\[ PA^2 - PB^2 = S_1 - S_2 \]
This can be simplified to:
\[ S_1 - S_2 = (2g_1 - 2g_2)x + (2f_1 - 2f_2)y + (c_1 - c_2) \]

4. **The radical axis of the two circles is defined by the equation:**
\[ S_1 - S_2 = 0 \]
This means that the radical axis is the locus of points where the tangents from P to both circles are equal.

5. **Let the perpendicular distance from point P to the radical axis be denoted as d.** The relationship between the difference of the squares of the tangents and the distance d can be expressed as:
\[ PA^2 - PB^2 = k \cdot d \]
where k is a constant that depends on the circles.

### Part 2: Prove that the locus of points such that the difference of the squares of the tangents from it to the two given circles is constant is a line parallel to their radical axis.

1. **Assume the difference of the squares of the tangents is a constant, say C:**
\[ PA^2 - PB^2 = C \]

2. **From the previous part, we know:**
\[ S_1 - S_2 = (2g_1 - 2g_2)x + (2f_1 - 2f_2)y + (c_1 - c_2) = C \]

3. **Rearranging gives us a linear equation in x and y:**
\[ (2g_1 - 2g_2)x + (2f_1 - 2f_2)y = C - (c_1 - c_2) \]

4. **This equation represents a straight line in the xy-plane.** Since the coefficients of x and y are constants, this line is parallel to the radical axis, which is defined by the equation \( S_1 - S_2 = 0 \).

### Conclusion:
We have shown that the difference of the squares of the tangents to the two circles from any point P varies as the perpendicular from P on their radical axis. Additionally, we have proven that the locus of points where the difference of the squares of the tangents is constant is a line parallel to the radical axis.