Answer:
# The general form of x²/3² + (y - 2)²/2² = 1 is 4x² - 36y + 9y² = 0
# The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0
# The general form of x²/6² + (y + 4)²/4² = 1 is 16x² + 288y + 36y² = 0
# The general form of (x - 3)²/3² + y²/5² = 1 is 25x² - 150x + 9y² = 0
Explanation:
* Lets revise the general form and the standard form of the ellipse
- The general form is Ax² + Bxy + Cy² + Dx + Ey + F = 0
- The standard form is (x - h)²/a² + (y - k)²/b² = 1
* Lets solve the problem
# x²/3² + (y - 2)²/2² = 1
∵ x²/9 + (y - 2)²/4 = 1 ⇒ multiply both sides by 9 × 4 (36)
∴ 4x² + 9(y - 2)² = 36 ⇒ open the bracket power 2
∴ 4x² + 9(y² - 4y + 4) = 36 ⇒ multiply the bracket by 9
∴ 4x² + 9y² - 36y + 36 = 36 ⇒ subtract 36 from both sides
∴ 4x² - 36y + 9y² = 0
* The general form of x²/3² + (y - 2)²/2² = 1 is 4x² + 9y² - 36y = 0
# (x + 7)²/7² + y²/6² = 1
∵ (x + 7)²/49 + y²/36 = 1 ⇒ multiply both sides by 49 × 36 (1764)
∴ 36(x + 7)² + 49y² = 1764 ⇒ open the bracket power 2
∴ 36(x² + 14x + 49) + 49y² = 1764 ⇒ multiply the bracket by 36
∴ 36x² + 504x + 1764 + 49y² = 1764 ⇒ subtract 1764 from both sides
∴ 36x² + 504x + 49y² = 0
* The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0
# x²/6² + (y + 4)²/4² = 1
∵ x²/36 + (y + 4)²/16 = 1 ⇒ multiply both sides by 36 × 16 (576)
∴ 16x² + 36(y + 4)² = 576 ⇒ open the bracket power 2
∴ 16x² + 36(y² + 8y + 16) = 576 ⇒ multiply the bracket by 36
∴ 16x² + 36y² + 288y + 576 = 576 ⇒ subtract 576 from both sides
∴ 16x² + 288y + 36y² = 0
* The general form of x²/6² + (y + 4)²/4² = 1 is 16x² + 288y + 36y² = 0
# (x - 3)²/3² + y²/5² = 1
∵ (x - 3)²/9 + y²/25 = 1 ⇒ multiply both sides by 9 × 25 (225)
∴ 25(x - 3)² + 9y² = 225 ⇒ open the bracket power 2
∴ 25(x² - 6x + 9) + 9y² = 225 ⇒ multiply the bracket by 25
∴ 25x² - 150x + 225 + 9y² = 225 ⇒ subtract 225 from both sides
∴ 25x² - 150x + 9y² = 0
* The general form of (x - 3)²/3² + y²/5² = 1 is 25x² - 150x + 9y² = 0