Answer:
(x - 2)²/2² + (y + 1)²/3² = 1 ⇒ The bold values and signs are the answers
Explanation:
* Lets revise the equation of the ellipse
- The standard form of the equation of an ellipse with center (h , k)
and major axis parallel to x-axis is (x - h)²/a² + (y - k)²/b² = 1
- The coordinates of the vertices are (h ± a , k)
- To change the form of the equation of the ellipse to standard form we
will using the completing square
∵ The equation of the ellipse is 9x² + 4y² - 36x + 8y + 4 = 0
- Lets collect x in bracket and y in bracket
∴ (9x² - 36x) + (4y² + 8y) + 4 = 0
- We will take a common factor 9 from the bracket of x and 4 from the
bracket of y
∴ 9(x² - 4x) + 4(y² + 2y) + 4 = 0
- Lets make 9(x² - 4x) a completing square
∵ √x² = x ⇒ the 1st term in the bracket
∵ 4x ÷ 2 = 2x ⇒ the product of the 1st and 2nd terms
∵ 2x ÷ x = 2 ⇒ the 2nd term in the bracket
∴ The bracket is (x - 2)²
∵ (x - 2)² = x² - 4x + 4 ⇒ we will add 4 in the bracket and subtract 4
out the bracket
∴ 9[(x² - 4x + 4) - 4] = 9[(x - 2)² - 4]
- Lets make 4(y² + 2y) a completing square
∵ √y² = y ⇒ the 1st term in the bracket
∵ 2y ÷ 2 = y ⇒ the product of the 1st and 2nd terms
∵ y ÷ y = 1 ⇒ the 2nd term in the bracket
∴ The bracket is (y + 1)²
∵ (y + 1)² = y² - 2y + 1 ⇒ we will add 1 in the bracket and subtract 1
out the bracket
∴ 4[(y² + 2y + 1) - 1] = 4[(y + 1)² - 1]
- Lets write the equation with the completing square
∴ 9[(x - 2)² - 4] + 4[(y + 1)² - 1] + 4 = 0 ⇒ simplify
∴ 9(x -2)² - 36 + 4(y + 1)² - 4 + 4 = 0 ⇒ add the numerical terms
∴ 9(x - 2)² + 4(y + 1)² - 36 = 0 ⇒ add 36 to both sides
∴ 9(x - 2)² + 4(y + 1)² = 36 ⇒ divide both sides by 36
∴ (x - 2)²/4 + (y + 1)²/9 = 1
∵ 4 = 2² and 9 = 3²
∴ (x - 2)²/2² + (y + 1)²/3² = 1
* The standard form of the equation of the ellipse is
(x - 2)²/2² + (y + 1)²/3² = 1