How do i write a equation of 8x+x^2-2y=64-y^2 ?
I recognize this to be an circle.
Do you want it in standard form?
if so, then ...
8x+x^2-2y=64-y^2
x^2 + 8x + ... + y^2 - 2y + ... = 64
complete the square
x^2 + 8x + 16 + y^2 - 2y + 1 = 64 + 16 + 1
(x + 4)^2 + (y-1)^2 = 81
centre is (-4,1), radius is 9
To write the equation 8x + x^2 - 2y = 64 - y^2 in a specific form, we need to rearrange the terms and simplify as much as possible. Let's go through the steps together:
Step 1: Combine like terms.
8x + x^2 - 2y = 64 - y^2
Step 2: Move all terms to one side to make it equal to zero.
x^2 + 8x + y^2 - 2y - 64 = 0
Step 3: Rearrange the terms in standard form.
x^2 + 8x + y^2 - 2y - 64 = 0
Now let's take a look at how to write the equation in a different form, specifically in the form of a circle equation.
Step 1: Group the x-terms together and the y-terms together.
(x^2 + 8x) + (y^2 - 2y) = 64
Step 2: Complete the square for both the x-terms and y-terms individually.
For the x-terms:
- Take half of the coefficient of 'x' (which is 8) and square it: (8/2)^2 = 16.
- Add this value inside the parentheses and subtract it outside the parentheses.
(x^2 + 8x + 16) - 16 + (y^2 - 2y) = 64
For the y-terms:
- Take half of the coefficient of 'y' (which is -2) and square it: (-2/2)^2 = 1.
- Add this value inside the parentheses and subtract it outside the parentheses.
(x^2 + 8x + 16) - 16 + (y^2 - 2y + 1) - 1 = 64
Step 3: Simplify the equation after completing the square.
(x + 4)^2 + (y - 1)^2 - 16 - 1 = 64
(x + 4)^2 + (y - 1)^2 - 17 = 64
Step 4: Move the constant terms to the other side of the equation.
(x + 4)^2 + (y - 1)^2 = 64 + 17
(x + 4)^2 + (y - 1)^2 = 81
Now the equation 8x + x^2 - 2y = 64 - y^2 is written in the form of a circle equation: (x + 4)^2 + (y - 1)^2 = 81.