Example 1:
Determine whether the circles defined by $x^2 + y^2 = 25$ and $(x-7)^2 + y^2 = 9$ touch externally or internally. Solution:
- Circle 1: Center $(0, 0)$, Radius $5$ - Circle 2: Center $(7, 0)$, Radius $3$ - Distance $d = \sqrt{(7-0)^2 + (0-0)^2} = 7$ Check the sum and difference of radii: - $r_1 + r_2 = 5 + 3 = 8$ - $|r_1 - r_2| = |5 - 3| = 2$ Since $d = 7$ is neither equal to $8$ nor $2$, the circles intersect at two points. Example 2:
Determine whether the circles defined by $(x-2)^2 + (y-3)^2 = 16$ and $(x-2)^2 + (y-3)^2 = 9$ touch internally. Solution:
- Circle 1: Center $(2, 3)$, Radius $4$ - Circle 2: Center $(2, 3)$, Radius $3$ - Distance $d = \sqrt{(2-2)^2 + (3-3)^2} = 0$ Since one circle is entirely within the other and $d = |4 - 3| = 1$, but $d = 0 \neq 1$, the circles are concentric and do not touch internally.

Understanding and applying the correct equations is vital for solving problems related to the positions of two circles.

• Distance Between Centers: $d = \sqrt{(h_2 - h_1)^2 + (k_2 - k_1)^2}$
• Condition for Externally Touching: $d = r_1 + r_2$
• Condition for Internally Touching: $d = |r_1 - r_2|$

Key Points:

• If $d > r_1 + r_2$, the circles are separate and do not touch.
• If $d = r_1 + r_2$, the circles touch externally.
• If $|r_1 - r_2| < d < r_1 + r_2$, the circles intersect at two points.
• If $d = |r_1 - r_2|$, the circles touch internally.
• If $d < |r_1 - r_2|$, one circle lies entirely within the other without touching.

In coordinate geometry, determining the nature of the interaction between two circles involves analyzing their equations and the distance between their centers. Step-by-Step Approach:

1. Identify the centers and radii of both circles from their equations.
2. Calculate the distance $d$ between the centers using the distance formula.
3. Compare $d$ with the sum and difference of the radii to determine the nature of touching.

Example:
Determine whether the circles $$(x + 1)^2 + (y - 2)^2 = 16$$ and $$(x - 5)^2 + (y + 2)^2 = 9$$ touch externally or internally. Solution:
- Circle 1: Center $(-1, 2)$, Radius $4$ - Circle 2: Center $(5, -2)$, Radius $3$ - Distance $d = \sqrt{(5 - (-1))^2 + (-2 - 2)^2} = \sqrt{6^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21$ Compare with sum and difference of radii: - $r_1 + r_2 = 4 + 3 = 7$ - $|r_1 - r_2| = |4 - 3| = 1$ Since $d \approx 7.21 > 7$, the circles do not touch externally; they intersect at two points.

Delving deeper into the relationship between two circles, we explore the theoretical underpinnings that govern their interactions. Pitagoras' Theorem in Circle Geometry: In scenarios where two circles touch internally or externally, the Pythagorean theorem can be utilized to derive relationships between their radii and the distance between their centers. Proof of External Touching Condition:
Given two circles with centers $C_1(h_1, k_1)$ and $C_2(h_2, k_2)$, and radii $r_1$ and $r_2$ respectively. If they touch externally, the distance between the centers $d = r_1 + r_2$. By applying the distance formula: $$ d = \sqrt{(h_2 - h_1)^2 + (k_2 - k_1)^2} = r_1 + r_2 $$ Squaring both sides: $$ (h_2 - h_1)^2 + (k_2 - k_1)^2 = (r_1 + r_2)^2 $$ This establishes the condition for external touching. Proof of Internal Touching Condition:
If the circles touch internally, the distance between centers $d = |r_1 - r_2|$. Applying the distance formula: $$ d = \sqrt{(h_2 - h_1)^2 + (k_2 - k_1)^2} = |r_1 - r_2| $$ Squaring both sides: $$ (h_2 - h_1)^2 + (k_2 - k_1)^2 = (r_1 - r_2)^2 $$ This condition ensures internal touching.

Tackling complex problems involving the interaction of two circles often requires a multi-step approach, integrating various concepts of coordinate geometry. Problem 1:
Given two circles: $$ C_1: (x - 3)^2 + (y + 1)^2 = 25 $$ $$ C_2: (x + 2)^2 + (y - 3)^2 = 16 $$ Determine whether they touch externally, internally, or do not touch. Solution:
- Circle 1: Center $(3, -1)$, Radius $5$ - Circle 2: Center $(-2, 3)$, Radius $4$ - Distance $d = \sqrt{(3 - (-2))^2 + (-1 - 3)^2} = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.40$ Sum of radii: $5 + 4 = 9$ Difference of radii: $|5 - 4| = 1$ Since $1 < 6.40 < 9$, the circles intersect at two points and do not touch externally or internally. Problem 2:
Find the condition for two circles to be orthogonal (intersect at right angles) and determine if external or internal touching satisfies orthogonality. Solution:
Two circles are orthogonal if the square of the distance between their centers is equal to the sum of the squares of their radii: $$ d^2 = r_1^2 + r_2^2 $$ For external touching: $$ d = r_1 + r_2 \implies d^2 = (r_1 + r_2)^2 = r_1^2 + 2r_1r_2 + r_2^2 $$ This does not satisfy $d^2 = r_1^2 + r_2^2$ unless $r_1r_2 = 0$, which is not possible for valid circles. For internal touching: $$ d = |r_1 - r_2| \implies d^2 = (r_1 - r_2)^2 = r_1^2 - 2r_1r_2 + r_2^2 $$ This also does not satisfy $d^2 = r_1^2 + r_2^2$ unless $r_1r_2 = 0$. Therefore, external and internal touching do not satisfy the condition for orthogonality.

The concepts of circles touching externally or internally have applications beyond pure mathematics, influencing fields such as engineering, physics, and computer graphics.

• Engineering: Understanding the interactions between circular gears involves analyzing points of contact and ensuring proper meshing without slippage, often relying on the principles of external and internal tangency.
• Physics: In optics, the principles of circle interactions help in designing lenses and understanding light reflection and refraction at curved surfaces.
• Computer Graphics: Rendering circles and spheres accurately in digital environments requires precise calculations of their positions and interactions, including external and internal touching scenarios.
• Robotics: Path planning for circular robots or components often involves ensuring that parts move without unintended contact, using the principles of circle tangency.

Real-world problems often model situations using circles, where understanding their interactions is crucial. Urban Planning:
Designing circular roundabouts requires ensuring that entry and exit points are smoothly integrated without causing traffic congestion, analogous to externally touching circles. Mechanical Design:
Creating components like pulleys and bearings involves precise alignment, ensuring that circular parts touch internally or externally as needed to function correctly. Navigation Systems:
The concept of circles is fundamental in determining ranges of signals and coverage areas, requiring analysis of overlapping or touching zones.

• Determining whether two circles touch externally or internally relies on calculating the distance between their centers relative to their radii.
• Externally touching circles satisfy $d = r_1 + r_2$, while internally touching circles satisfy $d = |r_1 - r_2|$.
• Understanding these concepts is essential for solving complex geometric problems and has practical applications in various engineering and technological fields.
• Visualizing and applying the correct formulas ensures accurate analysis of circle interactions in coordinate geometry.