Circle Equation: Center, Radius, and Standard Form

A circle is the set of all points at a fixed distance $r$ (the radius) from a fixed point $(h, k)$ (the center). This geometric definition translates directly into an algebraic equation:

$(x - h)^{2} + (y - k)^{2} = r^{2}$

This is the standard form of a circle's equation. It's a direct application of the distance formula: the distance from any point $(x, y)$ on the circle to the center $(h, k)$ equals $r$.

Solving the problem

For $(x - 3)^{2} + (y - 2)^{2} = 25$:

Comparing with $(x - h)^{2} + (y - k)^{2} = r^{2}$:

• $h = 3$, $k = 2$ → center is $(3, 2)$
• $r^{2} = 25$ → $r = 5$

The circle is centered at $(3, 2)$ with radius $5$. It passes through $(8, 2)$, $(-2, 2)$, $(3, 7)$, and $(3, -3)$ — each exactly 5 units from the center.

Converting to general form

Expand the standard form:

$(x - 3)^{2} + (y - 2)^{2} = 25$ $x^{2} - 6x + 9 + y^{2} - 4y + 4 = 25$ $x^{2} + y^{2} - 6x - 4y - 12 = 0$

This is the general form: $x^{2} + y^{2} + Dx + Ey + F = 0$, where $D = -6$, $E = -4$, $F = -12$.

Converting back: completing the square

Given $x^{2} + y^{2} - 6x - 4y - 12 = 0$:

1. Group: $(x^{2} - 6x) + (y^{2} - 4y) = 12$
2. Complete each square: $(x^{2} - 6x + 9) + (y^{2} - 4y + 4) = 12 + 9 + 4$
3. Factor: $(x - 3)^{2} + (y - 2)^{2} = 25$

The center is $(-D/2, -E/2) = (3, 2)$ and $r = \sqrt{(D/2)^{2} + (E/2)^{2} - F} = \sqrt{9 + 4 + 12} = 5$.

Key properties

• Every point on the circle satisfies the equation; every point inside has distance $< r$; every point outside has distance $> r$.
• A line can intersect a circle at 0 (outside), 1 (tangent), or 2 points (secant).
• The circle with center at the origin simplifies to $x^{2} + y^{2} = r^{2}$ — the Pythagorean theorem in disguise.

Real-world applications

• GPS: Each satellite defines a sphere (3D circle) of possible positions; the intersection of 3+ spheres gives your location.
• Radar: Detection range is circular. Overlapping radar coverage is a union of circles.
• Engineering: Wheels, gears, pipes, and all round objects are circles.

Try it in the visualization

Drag the $h$, $k$, and $r$ sliders to move and resize the circle. Toggle "show center" and "show radius" to see the key features. Turn on "show general form" to see the equation update in both forms simultaneously. Toggle "show point test" to check whether a specific point is inside, on, or outside the circle.