What is the centre of the equation 4x^2 9y^2 - 32x 36y 11
There is addition sign b/n 4x^2 + 9y^2
I think you mean the center of the ellipse whose equation is (assuming the blanks represent + signs):
4x^2 + 9y^2 - 32x + 36y + 11 = 0
4x^2-32x + 9y^2+36y + 11 = 0
4(x^2-8x) + 9(y^2+4y) = -11
now complete the squares and add the same amounts on the right side:
4(x^2-8x+16) + 9(y^2+4y+4) = -11+4(16)+9(4)
4(x-4)^2 + 9(y+2)^2 = 89
center is at (4,-2)
If I got the original equation wrong, fix it and follow the same steps.
To find the center of the equation 4x^2 + 9y^2 - 32x + 36y = 11, we can follow a few steps:
Step 1: Rewrite the equation in standard form.
In order to find the center, we need to rewrite the equation in standard form, which is (x - h)^2 + (y - k)^2 = r^2. To do this, we need to complete the square for both the x and y terms.
Starting with the x terms:
4x^2 - 32x + 9y^2 + 36y = 11
To complete the square for the x terms, we need to take half of the coefficient of x, square it, and then add it to both sides of the equation. The coefficient of x is -32, so half of it is -16. Squaring -16 gives us 256, so we add it to both sides:
4x^2 - 32x + 256 + 9y^2 + 36y = 11 + 256
Simplifying the right side of the equation:
4x^2 - 32x + 9y^2 + 36y = 267
Now let's complete the square for the y terms:
To complete the square for the y terms, we need to take half of the coefficient of y, square it, and add it to both sides. The coefficient of y is 36, so half of it is 18. Squaring 18 gives us 324, so we add it to both sides:
4x^2 - 32x + 9y^2 + 36y + 324 = 267 + 324
Simplifying the right side of the equation:
4x^2 - 32x + 9y^2 + 36y + 324 = 591
Now we have the equation in standard form:
4x^2 - 32x + 9y^2 + 36y + 324 = 591
Step 2: Group like terms.
We can group the x terms and the y terms together to see the pattern:
(4x^2 - 32x) + (9y^2 + 36y) + 324 = 591
Step 3: Complete the square for each variable separately.
Now, we will complete the square for the x terms:
4(x^2 - 8x) + (9y^2 + 36y) + 324 = 591
To complete the square for the x terms, we need to take half of the coefficient of x, square it, and then add it inside the parentheses. The coefficient of x is -8, so half of it is -4. Squaring -4 gives us 16, so we add it inside the parentheses:
4(x^2 - 8x + 16) + (9y^2 + 36y) + 324 = 591 + 4(16)
Simplifying the right side of the equation:
4(x^2 - 8x + 16) + (9y^2 + 36y) + 324 = 655
Now, let's complete the square for the y terms:
4(x^2 - 8x + 16) + 9(y^2 + 4y) + 324 = 655
To complete the square for the y terms, we need to take half of the coefficient of y, square it, and then add it inside the parentheses. The coefficient of y is 4, so half of it is 2. Squaring 2 gives us 4, so we add it inside the parentheses:
4(x^2 - 8x + 16) + 9(y^2 + 4y + 4) + 324 = 655 + 9(4)
Simplifying the right side of the equation:
4(x^2 - 8x + 16) + 9(y^2 + 4y + 4) + 324 = 691
Step 4: Simplify each perfect square trinomial.
Now, we simplify each perfect square trinomial inside the parentheses:
4(x - 4)^2 + 9(y + 2)^2 + 324 = 691
Step 5: Move the constant term to the other side of the equation.
To isolate the constant term, we need to move it to the other side of the equation:
4(x - 4)^2 + 9(y + 2)^2 = 691 - 324
Simplifying the right side of the equation:
4(x - 4)^2 + 9(y + 2)^2 = 367
Step 6: Write the equation in standard form.
Finally, we have the equation in standard form, where (x - h)^2 is the x-coordinate of the center and (y - k)^2 is the y-coordinate of the center:
4(x - 4)^2 + 9(y + 2)^2 = 367
Therefore, the center of the equation 4x^2 + 9y^2 - 32x + 36y = 11 is at the point (4, -2).