Circle: Center-Radius Equation

Step 1: Identify the center coordinates and the radius.

The center coordinates are given in the problem. The radius must be determined from the diameter, which is 2 times the radius or $r=\frac{d}{2}$.

Center: (-2, 5) Given

$Radius=\frac{diameter}{2}$

$r=\frac{12}{2}\to r=6$

Step 2: Substitute the center coordinates and radius into the center-radius form.

Be careful to include negative signs when substituting the center coordinates.

${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$   Original

${\left[x-\left(-2\right)\right]}^2+{\left(y-5\right)}^2=6^2$ Sub.

${\left(x+2\right)}^2+{\left(y-5\right)}^2=36$ Simplify

Step 1: Write the center-radius equation for the circle.

Center: (0, 0) Given

Radius = 8  Given

${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ Original

${\left(x-0\right)}^2+{\left(y-0\right)}^2=8^2$ Sub.

$x^2+y^2=64$ Simplify

Step 2: Determine the x-coordinate associated with a y-coordinate of -3.

Substitute -3 in for y and solve for x.

$x^2+y^2=64$ Original

$x^2+{\left(-3\right)}^2=64$ Sub.

$x^2+9=64$   Square

$x^2=55$      Subtract

$x=\sqrt{55}\approx 7.4$   Square Root