How do I determine whether two circles intersect?

Answer 1

Let's assume the first circle with origin (x1, y1) and radius r1 and the second

circle with origin (x2, y2) and radius r2. The order of the cicles does not

matter.

We can calculate the distance between the two origins:

d = sqrt(x2-x1)^2 + (y2-y1)^2)

(sqrt is the square root operation)

If d = 0, and r1 = r2, then the cicles are entirely overlapping, they intersect

everywhere.

If d > r1 + r2, then the circles are too far apart, and there is no intersection.

If d < abs(r1-r2), then the circle with the smaller radius is inside the circle

with the larger radius, and there is no intersection. (abs is the absolute

value.)

If d = r1 + r2, or d = abs(r1-r2), then there is one intersection point.

Otherwise, there are two intersection points.

Answer 2

To determine whether two circles intersect, you need to compare the distance between their centers to the sum of their radii. Here's a step-by-step approach:

1. Know the equations of both circles:

Circle 1: (x - h₁)² + (y - k₁)² = r₁²

Circle 2: (x - h₂)² + (y - k₂)² = r₂²

Where (h₁, k₁) and (h₂, k₂) are the center coordinates, and r₁ and r₂ are the radii.

2. Calculate the distance between the centers:

Use the distance formula: d = √[(h₂ - h₁)² + (k₂ - k₁)²]

3. Compare this distance to the sum and difference of the radii:

- If d > r₁ + r₂: The circles are separate.

- If d < |r₁ - r₂|: One circle is inside the other without touching.

- If d = r₁ + r₂: The circles are touching at one point (externally tangent).

- If d = |r₁ - r₂|: The circles are touching at one point (internally tangent).

- If |r₁ - r₂| < d < r₁ + r₂: The circles intersect at two points.

- If d = 0 and r₁ = r₂: The circles are coincident (same circle).

Example:

Circle 1: (x - 2)² + (y - 3)² = 4²

Circle 2: (x + 1)² + (y - 1)² = 3²

1. Centers: (2, 3) and (-1, 1)

2. d = √[(2 - (-1))² + (3 - 1)²] = √(3² + 2²) = √13 ≈ 3.61

3. r₁ + r₂ = 4 + 3 = 7

|r₁ - r₂| = |4 - 3| = 1

Since 1 < 3.61 < 7, these circles intersect at two points.