Question

determine the equation of the circle graphed below.

Explanation:

Step1: Recall circle equation formula

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

Step2: Identify the center

From the graph, the circle passes through the origin \((0,0)\) and its topmost point is \((0,10)\), and it touches the \(x\)-axis at \((0,0)\). The center \((h, k)\) should be at \((-2, 5)\)? Wait, no, let's check again. Wait, the circle's bottom point is at \((0,0)\) and top at \((0,10)\)? Wait, no, looking at the grid, the center's \(y\)-coordinate: the distance from center to bottom (on \(x\)-axis) and to top. Wait, the circle touches the \(x\)-axis at \((0,0)\) and the top is at \((0,10)\)? Wait, no, the center should be halfway between the bottom (on \(x\)-axis, \(y = 0\)) and the top ( \(y = 10\))? Wait, no, the center's \(y\)-coordinate: the vertical distance from center to \(x\)-axis (where the circle touches) is the radius. Wait, the circle touches the \(x\)-axis at \((0,0)\) and the center is at \((h, k)\). Let's find the center. Looking at the graph, the circle is symmetric. Let's see the horizontal and vertical positions. Wait, the circle's leftmost and rightmost? Wait, no, the center: from the graph, the center is at \((-2, 5)\)? Wait, no, let's count the grid. Wait, the circle touches the \(x\)-axis at \((0,0)\) and the top point is at \((0,10)\)? Wait, no, the \(y\)-axis at the top is 10? Wait, the grid lines: each square is 1 unit. So the center: the distance from center to \(x\)-axis (where the circle is tangent) is the radius. The top of the circle is at \(y = 10\), so the radius is 5? Wait, no, if the center is at \((h, k)\), and the circle touches the \(x\)-axis, then \(k = r\) (since the distance from center \((h,k)\) to \(x\)-axis ( \(y=0\)) is \(k\), which equals the radius \(r\)). Also, the top of the circle is at \(y = k + r = 2r\) (since \(k = r\)). From the graph, the top is at \(y = 10\), so \(2r = 10\) ⇒ \(r = 5\), so \(k = 5\). Now the \(x\)-coordinate of the center: looking at the graph, the circle is shifted left? Wait, the circle's center: let's see the horizontal position. Wait, the circle passes through \((0,0)\) and the center is \((h, 5)\). The distance from center \((h,5)\) to \((0,0)\) is the radius (5). So \(\sqrt{(0 - h)^2 + (0 - 5)^2} = 5\). So \((-h)^2 + 25 = 25\) ⇒ \(h^2 = 0\) ⇒ \(h = 0\)? Wait, that can't be. Wait, maybe I made a mistake. Wait, the circle touches the \(x\)-axis at \((0,0)\), so the distance from center \((h,k)\) to \((0,0)\) is equal to the radius, and also the distance from center to the top ( \(y = 10\)) is the radius. So \(k - 0 = r\) (distance from center to \(x\)-axis) and \(10 - k = r\) (distance from center to top). So \(k = r\) and \(10 - k = r\) ⇒ \(k = 10 - k\) ⇒ \(2k = 10\) ⇒ \(k = 5\), so \(r = 5\). Now, the \(x\)-coordinate: the center is at \((h, 5)\). Let's find \(h\). Looking at the graph, the circle is symmetric about the vertical line \(x = -2\)? Wait, no, maybe I misread. Wait, the circle's leftmost point: let's count the grid. Wait, the center is at \((-2, 5)\)? Wait, no, let's check the horizontal distance. Wait, the circle passes through \((0,0)\) and the center is \((h, 5)\). So the distance from \((h,5)\) to \((0,0)\) is 5 (radius). So \(\sqrt{(0 - h)^2 + (0 - 5)^2} = 5\) ⇒ \(\sqrt{h^2 + 25} = 5\) ⇒ \(h^2 + 25 = 25\) ⇒ \(h^2 = 0\) ⇒ \(h = 0\). Wait, that would mean center is \((0,5)\), radius 5. Let's check: \((x - 0)^2 + (y - 5)^2 = 5^2\) ⇒ \(x^2 + (y - 5)^2 = 25\). Let's see if this fits. When \(y = 0\), \(x^2 + 25 = 25\) ⇒ \(x^2 = 0\) ⇒ \(x = 0\), which matches the graph (touches \(x\)-axis at \((0,0)\)). When \(x = 0\), \((y - 5)^2 = 25\) ⇒ \(y - 5 = \pm5\) ⇒ \(y = 10\) or \(y = 0\), which matches the top at \((0,10)\) and bottom at \((0,0)\). Wait, but the graph looks shifted left? Wait, maybe I made a mistake in the center's \(x\)-coordinate. Wait, looking at the graph, the circle is not centered on the \(y\)-axis? Wait, the left side of the circle: let's count the grid. The circle's leftmost point: from center \((0,5)\), radius 5, so leftmost \(x\) is \(0 - 5 = -5\), rightmost \(x = 0 + 5 = 5\). Let's check the graph: the circle goes from \(x = -5\) to \(x = 5\)? Wait, the graph shows the circle on the grid, with the left side around \(x = -5\) and right side \(x = 5\), and top \(y = 10\), bottom \(y = 0\) (touching \(x\)-axis at \((0,0)\)). Wait, maybe my initial thought was wrong. So center is \((0,5)\), radius 5. So equation is \(x^2 + (y - 5)^2 = 25\). Wait, but let's check again. Wait, the circle in the graph: the center is at \((-2, 5)\)? No, the grid lines: each square is 1 unit. Let's look at the center's \(x\)-coordinate. The circle touches the \(x\)-axis at \((0,0)\) and the center is at \((h, 5)\). The horizontal distance from center to \((0,0)\) is \(|h - 0| = |h|\), and vertical distance is 5, so by Pythagoras, \(h^2 + 5^2 = 5^2\) ⇒ \(h = 0\). So center is \((0,5)\), radius 5. So equation is \(x^2 + (y - 5)^2 = 25\).

Step3: Confirm the radius and center

Center \((h, k) = (0, 5)\), radius \(r = 5\) (since distance from center to \(x\)-axis ( \(y=0\)) is 5, and to top \(y=10\) is also 5).

Step4: Write the equation

Substitute \(h = 0\), \(k = 5\), \(r = 5\) into the standard equation: \((x - 0)^2 + (y - 5)^2 = 5^2\), which simplifies to \(x^2 + (y - 5)^2 = 25\).

Answer:

\(x^2 + (y - 5)^2 = 25\)