What is the distance between the centers of two circles with equations (x - 1)²

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Question: What is the distance between the centers of two circles with equations (x - 1)² + (y - 2)² = 9 and (x + 3)² + (y + 4)² = 16?

Options:

Correct Answer: 6

Solution:

The distance between centers (1, 2) and (-3, -4) is calculated using the distance formula.

What is the distance between the centers of two circles with equations (x - 1)² + (y - 2)² = 9 and (x + 3)² + (y + 4)² = 16?

What is the distance between the centers of two circles with equations (x - 1)² + (y - 2)² = 9 and (x + 3)² + (y + 4)² = 16?

Step 1: Identify the center of the first circle from its equation (x - 1)² + (y - 2)² = 9. The center is (1, 2).
Step 2: Identify the center of the second circle from its equation (x + 3)² + (y + 4)² = 16. The center is (-3, -4).
Step 3: Use the distance formula to find the distance between the two centers. The distance formula is: distance = √((x2 - x1)² + (y2 - y1)²).
Step 4: Substitute the coordinates of the centers into the distance formula. Here, (x1, y1) = (1, 2) and (x2, y2) = (-3, -4).
Step 5: Calculate (x2 - x1) = (-3 - 1) = -4 and (y2 - y1) = (-4 - 2) = -6.
Step 6: Now calculate the squares: (-4)² = 16 and (-6)² = 36.
Step 7: Add the squares: 16 + 36 = 52.
Step 8: Take the square root of 52 to find the distance: √52 = 2√13.

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