9x²+4y²-18x+16y-11=0
9 x^2 -18 x = -4 y^2 -16 y + 11
x^2 - 2 x = -(4/9) y^2 -(16/9) y + 11/9
x^2 - 2 x + 1 =-(4/9) y^2 -(16/9) y+ 20/9
-(9/4)(x-1)^2 = y^2 + 4 y - 5
-(9/4)(x-1)^2 + 5 + 4 = y^2 + 4 y +4
-(9/4)(x-1)^2 + 9 = (y+2)^2
+ 9 = (9/4)(x-1)^2 + (y+2)^2
(x-1)^2/4 + (y+2)^2/9 = 1
ellipse center at (1,-2) semi axes 2 and 3
To solve the equation 9x² + 4y² - 18x + 16y - 11 = 0, you can follow these steps:
Step 1: Group the x and y terms separately:
(9x² - 18x) + (4y² + 16y) - 11 = 0
Step 2: Complete the square for the x terms inside the parentheses:
Take the coefficient of x, which is -18, divide it by 2, and square it,
(-18/2)² = 81.
Add 81 inside the first set of parentheses, but since we added 81, we also need to subtract it outside the parentheses to maintain the balance:
(9x² - 18x + 81) + (4y² + 16y) - 11 - 81 = 0
(9x² - 18x + 81) + (4y² + 16y) - 92 = 0
Step 3: Complete the square for the y terms inside the parentheses:
Take the coefficient of y, which is 16, divide it by 2, and square it,
(16/2)² = 64.
Add 64 inside the second set of parentheses, but since we added 64, we also need to subtract it outside the parentheses to maintain the balance:
(9x² - 18x + 81) + (4y² + 16y + 64) - 92 - 64 = 0
(9x² - 18x + 81) + (4y² + 16y + 64) - 156 = 0
Step 4: Factor the perfect square trinomials inside both sets of parentheses:
(3x - 9)² + (2y + 8)² - 156 = 0
Step 5: Combine the constants:
(3x - 9)² + (2y + 8)² = 156
Step 6: Divide both sides of the equation by 156 to isolate the squared terms:
[(3x - 9)²]/156 + [(2y + 8)²]/156 = 1
Step 7: Simplify the fractions if possible (optional):
[(x - 3/3)²]/(156/3) + [(y + 4)²]/(78/2) = 1
(x - 1/3)²/(52/3) + (y + 4)²/39 = 1
Therefore, the equation 9x² + 4y² - 18x + 16y - 11 = 0 rearranges to the standard form of an ellipse: (x - 1/3)²/(52/3) + (y + 4)²/39 = 1.
The given expression is a quadratic equation in two variables, x and y. To understand the equation and determine its shape, we can rewrite it using standard quadratic form:
9x² - 18x + 4y² + 16y - 11 = 0
Now, let's focus on solving the equation by completing the square. To do this, we need to rearrange the terms by grouping the x-terms together and the y-terms together:
(9x² - 18x) + (4y² + 16y) - 11 = 0
Now, let's complete the square for the x-terms. To do this, we need to take half the coefficient of x (-18) and square it, which gives us 9² = 81. We then add and subtract this value:
(9x² - 18x + 81) + (4y² + 16y) - 11 - 81 = 0
Simplifying further, we can write:
(9x² - 18x + 81) + (4y² + 16y + 64) - 92 = 0
Now, let's focus on completing the square for the y-terms. Half the coefficient of y (16) is 8, and squaring it gives us 8² = 64. Adding and subtracting this value, we can write:
(9x² - 18x + 81) + (4y² + 16y + 64) - 92 - 64 = 0
Further simplifying, we have:
(9x² - 18x + 81) + (4y² + 16y + 64) - 156 = 0
We can combine the constant terms on the left side:
(9x² - 18x + 4y² + 16y) + (81 + 64 - 156) = 0
(9x² - 18x + 4y² + 16y) - 11 = 0
Finally, we can rewrite the expression as a perfect square trinomial for both x and y:
9(x² - 2x) + 4(y² + 4y) - 11 = 0
Now, we can factor the perfect square trinomials:
9(x² - 2x + 1) + 4(y² + 4y + 4) - 11 = 0
Expanding and simplifying, we get:
9(x - 1)² + 4(y + 2)² - 11 = 0
This equation represents an ellipse in the xy-plane with its center at (1, -2) and horizontal and vertical radii of √(11/9) and √(11/4), respectively.