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! From (x - 7)² + (y + 2)² = 49, the center is (7, -2) and radius is 7, as in choice A. Choice B flips the signs without accounting for the opposites in the equation. 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! For points 6 units from (-1, 4), it's (x + 1)² + (y - 4)² = 36, which is choice C. Choice B has the signs wrong and uses radius squared as 36 but with incorrect center signs. 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 = 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 - 2x + 12y + 20 = 0, group and complete: (x² - 2x) + (y² + 12y) = -20, add 1 and 36 to both sides for (x - 1)² + (y + 6)² = 17. Choice A correctly completes the square to (x - 1)² + (y + 6)² = 17. Choice B flips the signs incorrectly, changing 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 = 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 tower at (3, 4) with radius 10, it's (x - 3)² + (y - 4)² = 100, which is choice C. Choice A has the signs flipped, incorrectly centering at (-3, -4). 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 = 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 + 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. 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 center (0,0) passing through (5,12), the radius is √(5² + 12²) = √169 = 13, so x² + y² = 169, choice B. Choice A uses 13 without squaring it for r², a common mistake. 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 = 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. 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 = 16, which is choice C. Choice A has the signs flipped for the center, which would incorrectly place it at (-2, 3) instead of watching the opposites in the equation. 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 = 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 - 4x + 8y = 5, group and complete: (x² - 4x) + (y² + 8y) = 5, add 4 and 16 to both sides for (x - 2)² + (y + 4)² = 25. Choice A correctly completes the square to (x - 2)² + (y + 4)² = 25. Choice C uses the correct form but forgets to square the radius properly, using 5 instead of 25. 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 a tower at (3, 4) with reach 10 units, it's (x - 3)² + (y - 4)² = 100. Choice C correctly derives the equation as (x-3)² + (y-4)² = 100. Something like choice A may forget to square the radius or mix up coordinates. 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! Awesome application to real-world scenarios—keep it up!