Help please i need to write 4x^2 -y^2 -8x -4y +16 =0 in standard form.4x^2 -8x - y^2 -4y=-16 so 4(x^2 -2x +1)-(y^2-4y+4= -16 -4 +4 so 4(x-1)^2/-16 -(y-2)^2/-16 =1 so (x-1)^2/-4 -(y-2)^2/-16 so if I multiply everything by -1 it will get rid of my (-) on the left and I'm left with (x-1)^2/4 -(y-2)^2/16= but then I have to multiply the right side to and I'm back to having a -1 My center is (0,2) and its suppose to be (0,-2) can you explain what I'm doing wrong I know it must have something to do with the subtraction sign and negatives??
To write the equation 4x^2 - y^2 - 8x - 4y + 16 = 0 in standard form, you need to complete the square separately for the x and y terms.
1. Group the x terms and the y terms: (4x^2 - 8x) - (y^2 + 4y) + 16 = 0.
2. Complete the square for the x terms: First, divide the coefficient of x by 2 and square it: (-8/2)^2 = 16. Add this value to both sides inside the parentheses: (4x^2 - 8x + 16) - (y^2 + 4y) + 16 = 16.
3. Complete the square for the y terms: First, divide the coefficient of y by 2 and square it: (4/2)^2 = 4. Add this value to both sides inside the parentheses: (4x^2 - 8x + 16) - (y^2 + 4y + 4) + 16 = 16 + 4.
4. Simplify the expression inside the parentheses: (4x^2 - 8x + 16) - (y^2 + 4y + 4) + 16 = 20.
5. Rearrange the terms and combine like terms: (4x^2 - y^2) - 8x - 4y + 28 = 0.
The equation is now in standard form, with the x^2 and y^2 terms separated. However, there seems to be an error in your attempted solution. Let's go over the steps again and address the issue with signs.
1. Group the x terms and the y terms: (4x^2 - 8x) - (y^2 + 4y) + 16 = 0.
2. Complete the square for the x terms: Divide the coefficient of x by 2 and square it: (-8/2)^2 = 16. Add this value to both sides inside the parentheses: (4x^2 - 8x + 16) - (y^2 + 4y) + 16 = 16.
3. Complete the square for the y terms: Divide the coefficient of y by 2 and square it: (4/2)^2 = 4. Add this value to both sides inside the parentheses: (4x^2 - 8x + 16) - (y^2 + 4y + 4) + 16 = 16 + 4.
4. Simplify the expression inside the parentheses: (4x^2 - 8x + 16) - (y^2 + 4y + 4) + 16 = 20.
5. Rearrange the terms and combine like terms: (4x^2 - y^2) - 8x - 4y + 28 = 0.
Therefore, the equation in standard form is (4x^2 - y^2) - 8x - 4y + 28 = 0.
Regarding the issue with the center point, the equation you obtained is correct. The center of an ellipse in standard form is generally represented as (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, the center would be (1, -2). Double-check your calculations and make sure you haven't made any sign errors in previous steps.