Answer:
# x² + y² - 4x + 12y - 20 = 0 ⇒ (x - 2)² + (y + 6)² = 60
# 3x² + 3y² + 12x + 18y - 15 = 0 ⇒ (x + 2)² + (y + 3)² = 18
# 2x² + 2y² - 24x - 16y - 8 = 0 ⇒ (x - 6)² + (y - 4)² = 56
# x² + y² + 2x - 12y - 9 = 0 ⇒ (x + 1)² + (y - 6)² = 46
Explanation:
* Lets study the problem to solve it
- Use the terms of x and y in the general form to find the standard form
∵ x² + y² - 4x + 12y - 20 = 0
- Use the term x term
∵ -4x ÷ 2 = -2x ⇒ x × -2
∴ (x - 2)²
- Use the term y term
∵ 12y ÷ 2 = 6y ⇒ y × 6
∴ (y + 6)²
∵ (-2)² + (6)² + 20 = 4 + 36 + 20 = 60
∴ x² + y² - 4x + 12y - 20 = 0 ⇒ (x - 2)² + (y + 6)² = 60
∵ x² + y² + 6x - 8y + 10 = 0
- Use the term x term
∵ 6x ÷ 2 = 3x ⇒ x × 3
∴ (x + 3)²
- Use the term y term
∵ -8y ÷ 2 = -4y ⇒ y × -4
∴ (y - 4)²
∵ (3)² + (-4)² - 10 = 9 + 16 - 10 = 5
∴ x² + y² + 6x - 8y + 10 = 0 ⇒ (x + 3)² + (y - 4)² = 5 ⇒ not in answer
∵ 3x² + 3y² + 12x + 18y - 15 = 0 ⇒ divide all terms by 3
∴ x² + y² + 4x + 6y - 5 = 0
- Use the term x term
∵ 4x ÷ 2 = 2x ⇒ x × 2
∴ (x + 2)²
- Use the term y term
∵ 6y ÷ 2 = 3y ⇒ y × 3
∴ (y + 3)²
∵ (2)² + (3)² + 5 = 4 + 9 + 5 = 18
∴ 3x² + 3y² + 12x + 18y - 15 = 0 ⇒ (x + 2)² + (y + 3)² = 18
∵ 5x² + 5y² - 10x + 20y - 30 = 0 ⇒ divide both sides by 5
∴ x² + y² - 2x + 4y - 6 = 0
- Use the term x term
∵ -2x ÷ 2 = -x ⇒ x × -1
∴ (x - 1)²
- Use the term y term
∵ 4y ÷ 2 = 2y ⇒ y × 2
∴ (y + 2)²
∵ (-1)² + (2)² + 6 = 1 + 4 + 6 = 11
∴ 5x² + 5y² - 10x + 20y - 30 = 0 ⇒ (x - 1)² + (y + 2)² = 11 ⇒ not in answer
∵ 2x² + 2y² - 24x - 16y - 8 = 0 ⇒ divide both sides by 2
∴ x² + y² - 12x - 8y - 4 = 0
- Use the term x term
∵ -12x ÷ 2 = -6x ⇒ x × -6
∴ (x - 6)²
- Use the term y term
∵ -8y ÷ 2 = -4y ⇒ y × -4
∴ (y - 4)²
∵ (-6)² + (-4)² + 4 = 36 + 16 + 4 = 56
∴ 2x² + 2y² - 24x - 16y - 8 = 0 ⇒ (x - 6)² + (y - 4)² = 56
∵ x² + y² + 2x - 12y - 9 = 0
- Use the term x term
∵ 2x ÷ 2 = x ⇒ x × 1
∴ (x + 1)²
- Use the term y term
∵ -12y ÷ 2 = -6y ⇒ y × -6
∴ (y - 6)²
∵ (1)² + (-6)² + 9 = 1 + 36 + 9 = 46
∴ x² + y² + 2x - 12y - 9 = 0 ⇒ (x + 1)² + (y - 6)² = 46