12.5 Circles in the Coordinate Plane
A circle in the coordinate plane is the set of all points located at the same distance from a fixed point. That fixed point is called the center, while the common distance is called the radius. On top of that, using the distance formula and the Cartesian coordinate system, circles can be represented by equations, graphed accurately, and analyzed through their centers, radii, intercepts, and extreme points. Understanding this connection between geometry and algebra is essential for solving problems involving circles, curved boundaries, and real-world locations.
The Standard Equation of a Circle
The standard equation for a circle is
[ (x-h)^2+(y-k)^2=r^2 ]
where:
- ((h,k)) is the center of the circle.
- (r) is the radius.
- (r^2) is the square of the radius.
- ((x,y)) represents any point on the circle.
This equation comes directly from the distance formula. The distance between the center ((h,k)) and a point ((x,y)) on the circle is
[ \sqrt{(x-h)^2+(y-k)^2}. ]
Because every point on the circle is exactly (r) units from the center, squaring both sides gives
[ (x-h)^2+(y-k)^2=r^2. ]
That's why, the standard equation provides a precise algebraic description of every point belonging to the circle Easy to understand, harder to ignore. But it adds up..
Example 1: Identifying the Center and Radius
Consider the equation
[ (x-4)^2+(y+3)^2=25. ]
To identify the center, rewrite (y+3) in the form (y-(-3)):
[ (x-4)^2+[y-(-3)]^2=25. ]
Now compare the equation with the standard form:
[ (x-h)^2+(y-k)^2=r^2. ]
Thus,
[ h=4,\qquad k=-3,\qquad r^2=25. ]
Since (r=\sqrt{25}=5), the circle has:
- Center: ((4,-3))
- Radius: (5)
A common source of confusion involves the signs inside the parentheses. Plus, in ((x-4)^2), the center’s (x)-coordinate is positive four. In ((y+3)^2), the center’s (y)-coordinate is negative three And it works..
Four Key Points on a Circle
The center is not usually located on the circle. To find four especially useful points, move the radius horizontally and vertically from the center Not complicated — just consistent..
For a circle with center ((h,k)) and radius (r), the key points are:
[ (h+r,k),\quad (h-r,k),\quad (h,k+r),\quad (h,k-r). ]
These points are often called the cardinal points of the circle. They make it possible to sketch a reasonably accurate graph before plotting additional points No workaround needed..
Here's one way to look at it: the circle
[ (x-4)^2+(y+3)^2=25 ]
has center ((4,-3)) and radius (5). Its four key points
