Finding the equation of a circle on a grid involves identifying its center and radius and then substituting these values into the standard equation form. The standard equation of a circle is given by:
$(x - h)^2 + (y - k)^2 = r^2$
Where:
- $(h, k)$ represents the coordinates of the center of the circle.
- $r$ represents the radius of the circle.
- $x$ and $y$ are the variables representing the coordinates of any point on the circle.
Steps to Find the Equation
To determine the equation of a circle when it's drawn on a grid, follow these steps:
Step 1: Locate the Center $(h,k)$
Identify the exact coordinates of the center point of the circle on the grid. This point is equidistant from all points on the circle's circumference. Read its x-coordinate (the horizontal value) and its y-coordinate (the vertical value) to get the values for $h$ and $k$.
Step 2: Determine the Radius $r$
Find the distance from the center $(h,k)$ to any point on the circle's edge. On a grid, the easiest way to do this is often by measuring horizontally or vertically from the center to where the circle crosses a grid line directly above, below, to the left, or to the right of the center. The number of units from the center to the edge along one of these lines is the radius, $r$.
Step 3: Write the Equation Using $(h,k)$ and $r$
Substitute the values you found for $h$, $k$, and $r$ into the standard equation:
$(x - h)^2 + (y - k)^2 = r^2$
It is important to correctly handle the signs of the center coordinates when placing them into the brackets. As highlighted in the reference, the coordinates of the center go into the brackets with the opposite sign.
- If the center's x-coordinate ($h$) is positive, it becomes $(x - h)$ in the equation. If $h$ is negative, it becomes $(x - (-|h|))$, which simplifies to $(x + |h|)$.
- Similarly, if the center's y-coordinate ($k$) is positive, it becomes $(y - k)$. If $k$ is negative, it becomes $(y - (-|k|))$, simplifying to $(y + |k|)$.
The reference explicitly illustrates this: "Center at x is two means the bracket is minus two and the center at y is minus three means the bracket is plus three that's all there is to it." This confirms that a center at $(2, -3)$ would lead to the terms $(x-2)$ and $(y+3)$ in the equation.
Finally, calculate $r^2$ by squaring the radius you found in Step 2.
Example
Let's say you find a circle on a grid with its center at $(1, -2)$ and a radius of 3 units.
- Center: $(h, k) = (1, -2)$
- Radius: $r = 3$
- Substitute and Calculate $r^2$:
- $h = 1$, so the x-term is $(x - 1)^2$.
- $k = -2$, so the y-term is $(y - (-2))^2$, which is $(y + 2)^2$.
- $r^2 = 3^2 = 9$.
Plugging these values into the standard equation gives:
$(x - 1)^2 + (y + 2)^2 = 9$
This is the equation of the circle on the grid.
