This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. A circle is defined as all points at a fixed distance (radius r) from a center point (h, k): using the distance formula, the distance from any point (x, y) on the circle to the center is $ \sqrt{(x - h)^2 + (y - k)^2} = r $. Squaring both sides eliminates the radical and gives the standard form $ (x - h)^2 + (y - k)^2 = r^2 $. This equation comes directly from the Pythagorean Theorem applied to the right triangle formed by the horizontal distance (x - h), vertical distance (y - k), and radius r as hypotenuse! For center (0, 0) and radius 3, it's simply $ x^2 + y^2 = 9 $, since $ (x - 0)^2 + (y - 0)^2 = 3^2 $. Choice C correctly derives the equation with r² = 9 for the origin-centered circle. A distractor like choice B might use 6 instead of 9, perhaps confusing radius with diameter, but remember to square the radius. Deriving from center and radius: (1) Write the distance from general point (x, y) to center (h, k) using distance formula: $ \sqrt{(x - h)^2 + (y - k)^2} $, (2) Set equal to radius r, (3) Square both sides to get $ (x - h)^2 + (y - k)^2 = r^2 $. That's it!

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. When a circle equation is in general form x² + y² + Dx + Ey + F = 0, we complete the square in both x and y to reveal the center and radius: group x-terms (x² + Dx) and complete the square by adding (D/2)², do the same for y-terms with (E/2)², then rearrange to get (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is (-D/2, -E/2) and radius is √[(D/2)² + (E/2)² - F]. It's completing the square twice, once for each variable! For x² + y² + 4x - 10y + 13 = 0, it becomes (x + 2)² + (y - 5)² = 16. Choice A correctly completes the square to (x+2)² + (y-5)² = 16. A common issue, as in choice B, is mishandling signs when rewriting the center. Completing the square for circles: (1) Group x-terms together and y-terms together: (x² + Dx) + (y² + Ey) = -F, (2) Complete square in x by adding (D/2)² to both sides, (3) Complete square in y by adding (E/2)² to both sides, (4) Factor: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F, (5) Read center as (-D/2, -E/2) and radius as √[right side]. Example: x² + y² + 6x - 8y = 0 → (x + 3)² + (y - 4)² = 9 + 16 = 25, so center (-3, 4), radius 5. Super progress—you're a pro at this now!

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. A circle is defined as all points at a fixed distance (radius r) from a center point (h, k): using the distance formula, the distance from any point (x, y) on the circle to the center is √[(x - h)² + (y - k)²] = r. Squaring both sides eliminates the radical and gives the standard form (x - h)² + (y - k)² = r². This equation comes directly from the Pythagorean Theorem applied to the right triangle formed by the horizontal distance (x - h), vertical distance (y - k), and radius r as hypotenuse! Here, points 5 units from (1, 2) satisfy (x - 1)² + (y - 2)² = 25. Choice C correctly derives the equation as (x-1)² + (y-2)² = 25. A distractor like choice A forgets to square the radius—it's r², not r! Deriving from center and radius: (1) Write the distance from general point (x, y) to center (h, k) using distance formula: √[(x - h)² + (y - k)²], (2) Set equal to radius r, (3) Square both sides to get (x - h)² + (y - k)² = r². That's it! For example, center (2, -5) and radius 6: (x - 2)² + (y - (-5))² = 6², which simplifies to (x - 2)² + (y + 5)² = 36. The Pythagorean Theorem makes circle equations! Fantastic— you're nailing the distance concept!

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. When a circle equation is in general form x² + y² + Dx + Ey + F = 0, we complete the square in both x and y to reveal the center and radius: group x-terms (x² + Dx) and complete the square by adding (D/2)², do the same for y-terms with (E/2)², then rearrange to get (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is (-D/2, -E/2) and radius is √[(D/2)² + (E/2)² - F]. It's completing the square twice, once for each variable! For x² + y² + 10x - 2y + 17 = 0, move 17: x² + 10x + y² - 2y = -17, complete x: (x + 5)² - 25, y: (y - 1)² - 1, so (x + 5)² + (y - 1)² = -17 + 25 + 1 = 9. Choice A correctly rewrites to standard form with center (-5, 1) and r² = 9. A common error, like in choice D, is miscalculating the right side to 25 instead of 9, but gently check the arithmetic: -17 + 25 + 1 is indeed 9. Completing the square for circles: (1) Group x-terms together and y-terms together: (x² + Dx) + (y² + Ey) = -F, (2) Complete square in x by adding (D/2)² to both sides, (3) Complete square in y by adding (E/2)² to both sides, (4) Factor: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F, (5) Read center as (-D/2, -E/2) and radius as √[right side]. Example: x² + y² + 6x - 8y = 0 → (x + 3)² + (y - 4)² = 9 + 16 = 25, so center (-3, 4), radius 5.

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. When a circle equation is in general form x² + y² + Dx + Ey + F = 0, we complete the square in both x and y to reveal the center and radius: group x-terms (x² + Dx) and complete the square by adding (D/2)², do the same for y-terms with (E/2)², then rearrange to get (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is (-D/2, -E/2) and radius is √[(D/2)² + (E/2)² - F]. It's completing the square twice, once for each variable! For x² + y² + 6x - 4y - 12 = 0, group and complete: (x² + 6x) + (y² - 4y) = 12, add 9 and 4 to both sides for (x + 3)² + (y - 2)² = 25, so center (-3, 2) and radius 5. Choice A correctly identifies the center and radius as (-3, 2) and 5. Choice B flips the signs on the center, perhaps by not negating D/2 and E/2 properly. Completing the square for circles: (1) Group x-terms together and y-terms together: (x² + Dx) + (y² + Ey) = -F, (2) Complete square in x by adding (D/2)² to both sides, (3) Complete square in y by adding (E/2)² to both sides, (4) Factor: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F, (5) Read center as (-D/2, -E/2) and radius as √[right side]. Example: x² + y² + 6x - 8y = 0 → (x + 3)² + (y - 4)² = 25, so center (-3, 4), radius 5.

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. When a circle equation is in general form x² + y² + Dx + Ey + F = 0, we complete the square in both x and y to reveal the center and radius: group x-terms (x² + Dx) and complete the square by adding (D/2)², do the same for y-terms with (E/2)², then rearrange to get (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is (-D/2, -E/2) and radius is √[(D/2)² + (E/2)² - F]. It's completing the square twice, once for each variable! For x² + y² + 6x - 4y + 9 = 0, move 9: x² + 6x + y² - 4y = -9, complete x: (x + 3)² - 9, y: (y - 2)² - 4, so (x + 3)² + (y - 2)² = -9 + 9 + 4 = 4, center (-3, 2), radius 2. Choice B correctly identifies the center (-3, 2) and radius 2 after completing the square. A mistake like in choice A might not flip the signs for the center, using (3, -2) instead, but remember center is (-D/2, -E/2). Reading center from standard form has a sign trap: in (x - h)² + (y - k)² = r², the center is (h, k), but the signs in the equation are OPPOSITE! From (x - 3)² + (y + 2)² = 16, the center is (3, -2) because (x - 3) has center x = 3, and (y + 2) = (y - (-2)) has center y = -2. Think: what values make each squared term equal zero? Those are your center coordinates. Don't just copy the numbers—flip the signs!

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. A circle is defined as all points at a fixed distance (radius r) from a center point (h, k): using the distance formula, the distance from any point (x, y) on the circle to the center is √[(x - h)² + (y - k)²] = r. Squaring both sides eliminates the radical and gives the standard form (x - h)² + (y - k)² = r². This equation comes directly from the Pythagorean Theorem applied to the right triangle formed by the horizontal distance (x - h), vertical distance (y - k), and radius r as hypotenuse! With center (-2, 5) and radius 7, we substitute h = -2, k = 5, and r = 7 into the standard form: (x - (-2))² + (y - 5)² = 7², which simplifies to (x + 2)² + (y - 5)² = 49. Choice C correctly derives the equation as (x + 2)² + (y - 5)² = 49. Choice B incorrectly has r² = 7 instead of r² = 49, forgetting to square the radius. Deriving from center and radius: (1) Write the distance from general point (x, y) to center (h, k) using distance formula: √[(x - h)² + (y - k)²], (2) Set equal to radius r, (3) Square both sides to get (x - h)² + (y - k)² = r². That's it! For example, center (2, -5) and radius 6: (x - 2)² + (y - (-5))² = 6², which simplifies to (x - 2)² + (y + 5)² = 36. The Pythagorean Theorem makes circle equations!

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. When a circle equation is in general form x² + y² + Dx + Ey + F = 0, we complete the square in both x and y to reveal the center and radius: group x-terms (x² + Dx) and complete the square by adding (D/2)², do the same for y-terms with (E/2)², then rearrange to get (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is (-D/2, -E/2) and radius is √[(D/2)² + (E/2)² - F]. It's completing the square twice, once for each variable! For x² + y² + 10x - 2y + 17 = 0, group and complete: (x² + 10x) + (y² - 2y) = -17, add 25 and 1 to both sides for (x + 5)² + (y - 1)² = 9. Choice A correctly completes the square to (x + 5)² + (y - 1)² = 9. Choice B flips the signs incorrectly, which would change the center to (5, -1) instead of watching the completing process. Completing the square for circles: (1) Group x-terms together and y-terms together: (x² + Dx) + (y² + Ey) = -F, (2) Complete square in x by adding (D/2)² to both sides, (3) Complete square in y by adding (E/2)² to both sides, (4) Factor: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F, (5) Read center as (-D/2, -E/2) and radius as √[right side]. Example: x² + y² + 6x - 8y = 0 → (x + 3)² + (y - 4)² = 25, so center (-3, 4), radius 5.

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. A circle is defined as all points at a fixed distance (radius r) from a center point (h, k): using the distance formula, the distance from any point (x, y) on the circle to the center is √[(x - h)² + (y - k)²] = r. Squaring both sides eliminates the radical and gives the standard form (x - h)² + (y - k)² = r². This equation comes directly from the Pythagorean Theorem applied to the right triangle formed by the horizontal distance (x - h), vertical distance (y - k), and radius r as hypotenuse! For the center at (2, -3) and radius 4, plug in to get (x - 2)² + (y - (-3))² = 4², which simplifies to (x - 2)² + (y + 3)² = 16. Choice B correctly derives the equation with the proper signs for the center and squared radius. A common mistake, like in choice D, is flipping the sign for y, resulting in (y - 3) instead of (y + 3), but remember to use (y - k) where k = -3 means +3. Deriving from center and radius: (1) Write the distance from general point (x, y) to center (h, k) using distance formula: √[(x - h)² + (y - k)²], (2) Set equal to radius r, (3) Square both sides to get (x - h)² + (y - k)² = r². That's it! Reading center from standard form has a sign trap: in (x - h)² + (y - k)² = r², the center is (h, k), but the signs in the equation are OPPOSITE!

This question tests your understanding of how to derive circle equations using the Pythagorean Theorem (distance formula) or find a circle's center and radius by completing the square. When a circle equation is in general form x² + y² + Dx + Ey + F = 0, we complete the square in both x and y to reveal the center and radius: group x-terms (x² + Dx) and complete the square by adding (D/2)², do the same for y-terms with (E/2)², then rearrange to get (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is (-D/2, -E/2) and radius is √[(D/2)² + (E/2)² - F]. It's completing the square twice, once for each variable! For x² + y² - 8x - 6y + 9 = 0, it yields (x - 4)² + (y - 3)² = 16, so center (4, 3) and radius 4. Choice A correctly identifies the center and radius as (4, 3) and 4. A distractor like choice B likely forgets to flip signs for the center formula. Reading center from standard form has a sign trap: in (x - h)² + (y - k)² = r², the center is (h, k), but the signs in the equation are OPPOSITE! From (x - 3)² + (y + 2)² = 16, the center is (3, -2) because (x - 3) has center x = 3, and (y + 2) = (y - (-2)) has center y = -2. Think: what values make each squared term equal zero? Those are your center coordinates. Don't just copy the numbers—flip the signs! You're excelling at extracting center and radius—bravo!