Math101 · Independent Ontario learning library
Math101Circle Equation
A circle with centre $(h,k)$ and radius $r>0$ has equation $(x-h)^2+(y-k)^2=r^2$. Every solution point is distance $r$ from the centre, so the equation is the distance formula with the square root removed.
Circle equations translate fixed-distance geometry into algebra, enabling coordinate proofs, intersections, loci, and analytic geometry.
Intuition and core definition
A circle with centre $(h,k)$ and radius $r>0$ has equation $(x-h)^2+(y-k)^2=r^2$. Every solution point is distance $r$ from the centre, so the equation is the distance formula with the square root removed.
Notation, language, and conditions
Signs reverse inside grouped coordinates: $(x+3)^2$ corresponds to centre coordinate $h=-3$. General form $x^2+y^2+Dx+Ey+F=0$ can be converted by completing the square. Equal coefficients on $x^2,y^2$ and no $xy$ term characterize an axis-aligned circle equation after scaling.
Why this idea matters
The standard circle equation records squared distance from a centre, with radius determining the common distance of every point on the curve.
A dependable method
- Read centre and radius from standard form, or group $x$ and $y$ terms in general form.
- Complete the square in each variable, adding the same quantities to both sides.
- Write the equation as two squared binomials equal to a positive number.
- Take the positive square root of the right side for radius.
- Substitute centre offsets or a claimed point to check.
