A circle is tangent to the x-axis at the point (4, 0). What is the equation of t

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Question: A circle is tangent to the x-axis at the point (4, 0). What is the equation of the circle if its radius is 3?

Options:

Correct Answer: (x - 4)² + (y + 3)² = 9

Solution:

The center of the circle is (4, 3) since it is 3 units above the tangent point (4, 0).

A circle is tangent to the x-axis at the point (4, 0). What is the equation of the circle if its radius is 3?

A circle is tangent to the x-axis at the point (4, 0). What is the equation of the circle if its radius is 3?

Step 1: Identify the tangent point on the x-axis, which is given as (4, 0).
Step 2: Understand that the radius of the circle is 3 units.
Step 3: Since the circle is tangent to the x-axis at (4, 0), the center of the circle must be directly above this point.
Step 4: Calculate the center of the circle by moving 3 units up from the tangent point (4, 0). This gives us the center at (4, 3).
Step 5: Use the center (4, 3) and the radius 3 to write the equation of the circle.
Step 6: The standard equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Step 7: Substitute h = 4, k = 3, and r = 3 into the equation: (x - 4)² + (y - 3)² = 3².
Step 8: Simplify the equation: (x - 4)² + (y - 3)² = 9.

Circle Geometry – Understanding the properties of circles, including the relationship between the center, radius, and points of tangency.
Coordinate Geometry – Using coordinates to determine the position of points and the equation of geometric shapes.