Write the equation in standard form for the circle x2+y2+6y–40=0.Submit

The given equation is x² + y² + 6y - 40 = 0.

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius.

To convert the given equation into standard form, we need to complete the square for the y terms.

First, rearrange the equation to group the y terms together:

x² + (y² + 6y) - 40 = 0

To complete the square, we take half of the coefficient of y, square it, and add it to both sides of the equation. Half of 6 is 3, and 3² is 9.

x² + (y² + 6y + 9) = 40 + 9

This simplifies to:

x² + (y + 3)² = 49

So, the equation of the circle in standard form is (x - 0)² + (y + 3)² = 49, which means the circle is centered at (0, -3) and has a radius of 7 (since √49 = 7).