What will be the center, and radius of a circle with this equation? x^2 +5x+8y−4=−y^2+9x

Question

I kinda know how to do it but I’m stuck.

Answer 1

x² + 5x + 8y - 4 = -y² + 9x

x² - 4x + y² + 8y = 4

Now complete the square.

Answer 2

x² + 5x + 8y − 4 = −y² + 9x [add y² to both sides, subtract 9x from both sides, add 4 to both sides to get:

x² - 4x + y² + 8y = 4 [Now, create an empty space after the "-4x" term and after the "8y" term so that you can create two trinomials like this:

(x² - 4x + ___ ) + (y² + 8y + ____ ) = 4 [To find the value to plug into the "blanks", you start with the coefficient of "-4x" term which is "-4". Divide it by two (it becomes "-2"). Then square it (it becomes "4"). "4" is the number that goes in the first blank. But, if you add "4" to the left side of the equation, you must also add "4" to the right side of the equation. To find what goes in the second blank, take the coefficient of "8y", which is "8". Divide it by two which makes it "4". Now square it (it becomes "16"). "16' is the number that goes in the second blank. But if you add "16" to the left side of the equation, you must also add "16" to the right side of the equation. So the equation becomes:

(x² - 4x + 4) + (y² + 8y + 16) = 4 + 4 + 16 [Now, write each trinomial as a square, meaning (x +/- a)² where the value of "a" is the number you got when you divided the coefficient by 2 (for "x" it was "-2" and for "y" it was "4"). Also, add all the numbers up on the right side of the equation. So the equation becomes:

(x - 2)² + (y + 4)² = 24

This is the now "standard form of a circle" which is typically written as:

(x - h)² + (y - k)² = r² where the center is (h, k) and the radius is "r".

You read the center from the equation we got except the x-value of the center is the OPPOSITE of what is grouped with the "x" (in other words, "+2") and the y-value of the center is the OPPOSITE of what is grouped with the "y" (in other words "-4") making the center (2, -4). The radius of the circle is the square root of the number we got meaning the radius is √(24) which can be simplified by writing it as √(4•6) and then taking the square root of the "4" to get r = 2√6