A circle is defined by the equation x² + y² - 10x + 6y + 25 = 0. What is the rad

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Question: A circle is defined by the equation x² + y² - 10x + 6y + 25 = 0. What is the radius of the circle?

Options:

Correct Answer: 5

Solution:

Rearranging gives (x - 5)² + (y + 3)² = 9, so the radius is √9 = 3.

A circle is defined by the equation x² + y² - 10x + 6y + 25 = 0. What is the radius of the circle?

A circle is defined by the equation x² + y² - 10x + 6y + 25 = 0. What is the radius of the circle?

Step 1: Start with the given equation of the circle: x² + y² - 10x + 6y + 25 = 0.
Step 2: Rearrange the equation to group x and y terms: x² - 10x + y² + 6y + 25 = 0.
Step 3: Move the constant (25) to the other side of the equation: x² - 10x + y² + 6y = -25.
Step 4: Complete the square for the x terms (x² - 10x). Take half of -10 (which is -5), square it (25), and add it: (x - 5)².
Step 5: Complete the square for the y terms (y² + 6y). Take half of 6 (which is 3), square it (9), and add it: (y + 3)².
Step 6: Add the completed squares to the equation: (x - 5)² + (y + 3)² = -25 + 25 + 9.
Step 7: Simplify the right side: (x - 5)² + (y + 3)² = 9.
Step 8: The equation is now in standard form (x - h)² + (y - k)² = r², where r is the radius.
Step 9: The radius r is the square root of 9, which is 3.

Circle Equation – Understanding the standard form of a circle's equation and how to convert from general form to standard form.
Completing the Square – The process of rearranging a quadratic equation to identify the center and radius of a circle.