Write the equation of circle O centered at origin that passes through (9,-2)

Circle B with center (0,-2) that passes through (-6,0)
>For circle B, is the radius 6 in this case? So equation would be x^2+(x+2)^2=36, correct?
If this is the case, how would I solve for circle O?

center at the origin means

x^2 + y^2 = r^2
9^2 + (-2)^2 = 81+4 = 85, so the circle is

x^2 + y^2 = 85

center at (0,-2) means
x^2 + (y+2)^2 = r^2
(-6)^2 + (0+2)^2 = 36+4 = 40

x^2 + (y+2)^2 = 40

Ah, solving for Circle O, the clowniest of circles! Well, since the center of Circle O is at the origin, that means the equation will be of the form x^2 + y^2 = r^2, where r is the radius.

We know that Point (9,-2) is on the circle. Substituting these coordinates into the equation, we get:

9^2 + (-2)^2 = r^2
81 + 4 = r^2
85 = r^2

So the equation for Circle O is x^2 + y^2 = 85. Ta-da!

For circle B, the equation would be (x - 0)^2 + (y - (-2))^2 = 6^2, which simplifies to x^2 + (y + 2)^2 = 36.

To find the equation of circle O, centered at the origin (0, 0) and passing through the point (9, -2), we can use the distance formula. The distance between the center (0, 0) and the point (9, -2) is equal to the radius of the circle.

Using the distance formula:
√((x2 - x1)^2 + (y2 - y1)^2) = radius

Substituting (x1, y1) = (0, 0) and (x2, y2) = (9, -2):
√((9 - 0)^2 + (-2 - 0)^2) = radius
√(9^2 + (-2)^2) = radius
√(81 + 4) = radius
√85 = radius

Therefore, the equation of circle O is x^2 + y^2 = 85.

To solve for the equation of circle O centered at the origin, we need to find its radius. The radius of a circle can be determined using the distance formula between the center of the circle and any point on the circumference.

In this case, the center of circle O is the origin (0,0), and it passes through the point (9,-2).

The distance formula is given by:

d = √[(x2 - x1)^2 + (y2 - y1)^2]

Let's plug in the values to find the distance between the origin and (9,-2):

d = √[(9 - 0)^2 + (-2 - 0)^2]
= √[(81 + 4)]
= √85

Therefore, the radius of circle O is √85.

Now we can write the equation of circle O. The equation of a circle centered at the origin is given by:

x^2 + y^2 = r^2

where r is the radius of the circle.

Substituting in the value of the radius (√85), the equation of circle O is:

x^2 + y^2 = (√85)^2
x^2 + y^2 = 85

So, the equation of circle O centered at the origin is x^2 + y^2 = 85.