The image provided explains the concept of the radical axis of two circles. I'll walk you through the explanation step by step.

Definition 5.1.1: Radical Axis of Two Circles

The radical axis of two circles is defined as the locus (set of points) where the lengths of tangents drawn from any point on the locus to the two circles are equal.

In simpler terms, if you take any point $P(x, y)$ on the radical axis and draw tangents from $P$ to both circles, the length of these tangents will be the same.

Equation of the Radical Axis

We are given two circles with their general equations:

1. $S = x^2 + y^2 + 2gx + 2fy + c = 0$ for Circle A
2. $S_1 = x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ for Circle B

We are asked to find the equation of the radical axis of these two circles.

Finding the Radical Axis

Let $P(x, y)$ be a point such that the lengths of tangents from $P$ to both circles are equal. The lengths of the tangents from $P$ to Circle A and Circle B are represented as:

• Tangent from $P$ to Circle A: $PT$
• Tangent from $P$ to Circle B: $PT_1$

Since the lengths of the tangents are equal, we have:

$PT = PT_1$ This implies: $PT^2 = PT_1^2$

Using the formula for the square of the length of a tangent from a point $P(x, y)$ to a circle $S$, we get:

$x^2 + y^2 + 2gx + 2fy + c = x^2 + y^2 + 2g_1x + 2f_1y + c_1$

Canceling out the $x^2$ and $y^2$ terms from both sides, we are left with:

$2(g - g_1)x + 2(f - f_1)y + (c - c_1) = 0$

Simplifying further:

$2(g - g_1)x + 2(f - f_1)y + (c - c_1) = 0$

This is the equation of the radical axis, which is a straight line. So, the radical axis of two given circles is always a straight line.

Geometrical Interpretation

In the diagram:

• $P(x, y)$ is a point on the radical axis.
• $T$ and $T_1$ are points where tangents touch circles A and B, respectively.
• Since $PT = PT_1$, point $P$ lies on the radical axis.

Thus, the radical axis is the locus of points from which the tangents to both circles are of equal length. Geometrically, this axis separates the two circles and contains points equidistant from the peripheries of the circles.

Key Insights

• The radical axis is always a straight line.
• It is formed by the set of points where tangents drawn to the two circles from any point on the line have equal length.

Would you like to explore more about any specific part of the explanation? Here are five related questions for further exploration:

1. How can you find the radical center when more than two circles are involved?
2. What happens to the radical axis if the two circles intersect?
3. Can the radical axis exist for circles that do not intersect or touch at all?
4. What is the significance of the radical axis in practical applications like geometry or physics?
5. How does the radical axis change if one of the circles shrinks to a point (i.e., becomes a point circle)?

Tip: The radical axis is perpendicular to the line joining the centers of the two circles. This property can help in visualizing and constructing it geometrically.