Ver en españolWrite the equation in standard form for the circle x2–12y+34=–y2.

The given equation is x^2 - 12y + 34 = -y^2.

Step 1: Move the y^2 term to the left side of the equation to combine like terms. This gives us x^2 + y^2 - 12y + 34 = 0.

Step 2: To write the equation in standard form for a circle, we need to complete the square for the y terms.

The standard form for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

Step 3: To complete the square, we take the coefficient of the y term, divide it by 2 and square it. (-12/2)^2 = 36.

Step 4: Add and subtract this inside the equation. This gives us x^2 + y^2 - 12y + 36 - 36 + 34 = 0.

Step 5: Rearrange the terms to get the completed square on one side: x^2 + (y - 6)^2 = 36 - 34 = 2.

So, the equation in standard form for the circle is x^2 + (y - 6)^2 = 2.