Question

complete the equation of this circle:

$(x - ?)^2 + (y - \quad)^2 = \quad$

Explanation:

Step1: Find the center coordinates

Looking at the grid, the center \( A \) seems to be at \( (4, 3) \) (by counting the grid squares from the origin; moving 4 units right on x - axis and 3 units up on y - axis). In the standard circle equation \((x - h)^2+(y - k)^2 = r^2\), \( (h,k) \) is the center. So \( h = 4 \), \( k=3 \).

Step2: Find the radius

We can find the radius by looking at the distance from the center to a point on the circle. Let's take the point where the circle intersects the y - axis (or we can count the grid squares). The distance from the center \( (4,3) \) to, say, the point \( (0,3) \) (if we assume the circle reaches x = 0) is \( 4 \) units? Wait, no, let's check the grid again. Wait, the circle passes through near the origin? Wait, maybe a better way: from the center \( (4,3) \), let's count the radius. Let's see, the circle seems to have a radius of 4? Wait, no, let's check the standard equation. Wait, maybe the center is at (4,3) and radius is 4? Wait, no, let's re - examine. Wait, the grid has 1 - unit squares. Let's find two points: center \( (h,k)=(4,3) \). Let's take a point on the circle, say, when x = 0, y = 3? No, maybe the circle intersects the x - axis? Wait, the circle passes through (0,0)? Wait, no, the graph shows the circle with center at (4,3) and radius 4? Wait, no, let's calculate the radius. 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.

Wait, maybe the center is at \( (4,3) \) and the radius is 4? Wait, no, let's count the distance from the center to the leftmost point. If the center is at \( x = 4 \), and the leftmost point is at \( x = 0 \), then the radius \( r=4 \). Similarly, the top - bottom distance: center at \( y = 3 \), if the bottom point is at \( y=-1 \), then the radius is 4? Wait, no, maybe I made a mistake. Wait, let's re - evaluate.

Wait, the center is at \( (4,3) \) (by counting: from the origin, 4 right, 3 up). Now, let's find the radius. Let's take the point where the circle intersects the x - axis. Wait, the circle passes through (0,0)? No, the graph shows the circle with center at (4,3) and radius 4? Wait, no, let's check the standard equation. Let's assume the center is \( (4,3) \) and radius \( r = 4 \). Then the equation is \((x - 4)^2+(y - 3)^2=16\)? Wait, no, \( r^2 = 16 \) if \( r = 4 \). Wait, maybe the radius is 4. Wait, let's confirm.

Wait, another way: the circle's center is at \( (h,k)=(4,3) \) and radius \( r = 4 \). So the equation is \((x - 4)^2+(y - 3)^2 = 16\). Wait, but let's check with the grid. If the center is at (4,3), and the radius is 4, then the circle will reach \( x=4 - 4=0 \), \( x = 4 + 4 = 8 \), \( y=3 - 4=-1 \), \( y=3 + 4 = 7 \). That seems to fit the graph.

So putting it into the equation: \((x - 4)^2+(y - 3)^2=16\)

Answer:

\((x - 4)^2+(y - 3)^2 = 16\) (So the values are 4, 3, 16)