Two Circles Two circles on a Cartesian plane, A and B, are each defined by three descriptors: • X: the x-coordinate of the circle’s center

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Two Circles Two circles on a Cartesian plane, A and B, are each defined by three descriptors: • X: the x-coordinate of the circle’s center • Y: the y-coordinate of the circle’s center • R: the radius of the circle Circles A and B will both be centered either on the X-axis (i.e. YA = 0 and YB = 0), or on the Y-axis (i.e. XA = 0 and XB = 0), but not both. A pair of circles (A and B) will have one of the following relationship types: • Touching: they touch each other at a single point • Concentric: they have the same center point • Intersecting: they intersect each other (touching at two points) • Disjoint-Outside: disjoint with one existing outside of the other • Disjoint-Inside: disjoint with one contained inside the other (but not concentric) Example: circlePairs = [’30 105 03′, ’01 40 15′] Each string has 6 integers that represent X, Y, and R for circle.

The Correct Answer and Explanation is:

✅ Answer: Disjoint-Inside

🧠 Explanation

We are given two circles defined as strings in the form 'X Y R', where:

  • X is the x-coordinate of the center
  • Y is the y-coordinate of the center
  • R is the radius

The two circles given are:

  1. '30 105 03' → Circle A: X=30, Y=105, R=3
  2. '01 40 15' → Circle B: X=1, Y=40, R=15

Let’s extract the values:

  • Circle A: center at (30, 105), radius = 3
  • Circle B: center at (1, 40), radius = 15

First, verify axis alignment:

  • Both circles have non-zero x and y, but to follow the problem constraints (centers lie either on the X-axis or Y-axis), we must assume either:
    • X-axis case: YA = 0 and YB = 0 → Not true (since YA = 105, YB = 40)
    • Y-axis case: XA = 0 and XB = 0 → Not true (since XA = 30, XB = 1)

However, the question may have a typo, or we assume the values were read with missing zeros. Possibly:

  • '30 105 03' = X=30, Y=0, R=105
  • '01 40 15' = X=1, Y=0, R=40

Let’s recalculate using this interpretation:

  • Circle A: center at (30, 0), radius = 105
  • Circle B: center at (1, 0), radius = 40

Now they are both centered on the X-axis, satisfying the condition.

Step 1: Distance between centers

  • Distance = |30 – 1| = 29

Step 2: Compare with radii

  • Radius A = 105
  • Radius B = 40
  • Sum = 145 → Distance (29) < 145
  • Difference = |105 – 40| = 65 → Distance (29) < 65

So, one circle is completely inside the other without touching.

✅ Relationship: Disjoint-Inside

Circle B is entirely inside Circle A, but their edges do not touch. Thus, they are disjoint inside.

Final Answer: Disjoint-Inside ✅

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