Question

write the equation of this circle in standard form.

Explanation:

Step1: Recall the standard form of a circle's equation

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

Step2: Identify the center of the circle

From the graph, the center of the circle is at \((0, -2)\), so \(h = 0\) and \(k = -2\).

Step3: Determine the radius of the circle

To find the radius, we can calculate the distance from the center \((0, -2)\) to a point on the circle, such as \((8, -2)\) (or \((0, 6)\), etc.). The distance between \((0, -2)\) and \((8, -2)\) is \(|8 - 0| = 8\)? Wait, no, wait. Wait, let's check another point. Wait, the center is \((0, -2)\), and a point on the circle is \((0, 6)\). The distance between \((0, -2)\) and \((0, 6)\) is \(|6 - (-2)| = 8\)? Wait, no, wait, the vertical distance is \(6 - (-2) = 8\)? Wait, but when we look at the horizontal distance, from center \((0, -2)\) to \((8, -2)\), the distance is \(8 - 0 = 8\). Wait, but let's check the graph again. Wait, the circle passes through \((8, -2)\) and \((0, 6)\). Wait, the center is \((0, -2)\), so the radius is the distance from \((0, -2)\) to \((8, -2)\), which is \(8\)? Wait, no, wait, the distance between \((0, -2)\) and \((8, -2)\) is \(8\) units (since the x-coordinates differ by 8, y-coordinates same). Alternatively, distance from \((0, -2)\) to \((0, 6)\) is \(6 - (-2) = 8\) units. So radius \(r = 8\)? Wait, no, wait, that can't be. Wait, let's count the grid. The center is at \((0, -2)\), and the circle reaches to \(x = 8\) (so from \(x = 0\) to \(x = 8\) is 8 units, so radius is 8? Wait, but when we check the top point, at \(y = 6\), center at \(y = -2\), so vertical distance is \(6 - (-2) = 8\), so radius is 8. Wait, but let's confirm. Wait, the standard form is \((x - h)^2 + (y - k)^2 = r^2\). So center \((h, k) = (0, -2)\), radius \(r = 8\)? Wait, no, wait, maybe I made a mistake. Wait, let's check the point \((8, -2)\): plug into the equation \((8 - 0)^2 + (-2 - (-2))^2 = 64 + 0 = 64\), so \(r^2 = 64\), so \(r = 8\). Alternatively, point \((0, 6)\): \((0 - 0)^2 + (6 - (-2))^2 = 0 + 64 = 64\), so \(r^2 = 64\), so \(r = 8\). So that's correct.

Wait, but let's re-express the standard form. So \(h = 0\), \(k = -2\), \(r = 8\). So the equation is \((x - 0)^2 + (y - (-2))^2 = 8^2\), which simplifies to \(x^2 + (y + 2)^2 = 64\). Wait, but let's check another point. Let's take \((-8, -2)\): \((-8 - 0)^2 + (-2 - (-2))^2 = 64 + 0 = 64\), which matches. And \((0, -10)\): \((0 - 0)^2 + (-10 - (-2))^2 = 0 + 64 = 64\), which also matches. So that's correct.

Wait, but initially, I thought maybe the radius was 5 or something, but no, the distance from center \((0, -2)\) to \((8, -2)\) is 8 units. So the radius is 8.

So putting it all together: center \((h, k) = (0, -2)\), radius \(r = 8\). So the standard form is \((x - 0)^2 + (y - (-2))^2 = 8^2\), which is \(x^2 + (y + 2)^2 = 64\).

Wait, but let's check the graph again. The center is at (0, -2), and the circle goes from x = -8 to x = 8 (since it passes through (-8, -2) and (8, -2)), so the diameter is 16, so radius is 8. Yes, that makes sense. So the equation is \(x^2 + (y + 2)^2 = 64\).

Answer:

The equation of the circle in standard form is \(\boxed{x^2 + (y + 2)^2 = 64}\)