ACT® · Math: Geometry — Coordinate Geometry and Conic Sections
The Equation of a Circle
A circle's equation is just the distance formula with the distance locked at a fixed value r. Every point (x, y) that satisfies (x - h)2 + (y - k)2 = r2 sits exactly r units from the center (h, k).
The distance formula says the distance between two points (x, y) and (h, k) is √((x - h)2 + (y - k)2). A circle is just the set of all points that are the same fixed distance, r, from one center point. So take that distance formula, set it equal to r, and square both sides to clear the square root. You get (x - h)2 + (y - k)2 = r2. Nothing mysterious is happening here beyond locking the distance down to a constant and squaring it so the equation has no radicals in it.
Reading the equation backward tells you the circle's whole story. Whatever is subtracted from x gives you h, the x-coordinate of the center, and whatever is subtracted from y gives you k, the y-coordinate. The number on the right is r squared, not r, so you take its square root to get the actual radius. Watch the signs carefully: the standard form is built with subtraction, so (x + 3)2 is really (x - (-3))2, meaning h = -3, not 3. This sign flip is the single most common place this topic gets missed.
This equation sits right next to the other coordinate geometry tools: distance, midpoint, and slope all feed into it, since the center is a point and the radius is a distance. It also opens the door to the other conic sections. A circle is the special case of an ellipse where both axes are equal, so once you're comfortable with squared terms adding up to a constant, the ellipse's equation with two different denominators will feel like a natural next step rather than a new idea.
Key terms
- Center
- The fixed point (h, k) that every point on the circle is the same distance from.
- Radius
- The constant distance r between the center and any point on the circle.
- Standard form
- The equation (x - h)2 + (y - k)2 = r2, written with subtraction so h and k read directly as the center.
- Distance formula
- The formula √((x2 - x1)2 + (y2 - y1)2) that the circle equation comes from after fixing the distance and squaring.
Watch it explained
- How it worksHow does (x - h)2 + (y - k)2 = r2 say that every point sits the same distance from the center?
- Worked exampleFind the center and radius of the circle (x + 3)2 + (y - 5)2 = 49.
- Common mistakeWhy isn't the center of (x + 3)2 + (y - 5)2 = 49 located at the point (3, -5)?
- Going furtherWhat happens to the equation when the circle's center slides onto the origin?
Check yourself
End of topic.