Circles & Parabolas as Conics

A circle with centre (h, k) and radius r has equation:

(x − h)² + (y − k)² = r²

For (x − 2)² + (y + 3)² = 25: centre (2, −3), radius 5. Watch the signs — y + 3 means k = −3 — and remember the right side is r², so the radius is √25 = 5.

If the equation is expanded instead, complete the square on both x and y to recover this form.

Parabolas are the other conic you meet in Algebra 2. y = a(x − h)² + k opens vertically; x = a(y − k)² + h opens horizontally. Swapping which variable is squared is what turns the parabola on its side.

Conic sections arise as intersections of a plane with a double cone, producing circles, ellipses, parabolas and hyperbolas. Their general second-degree equation is Ax² + Bxy + Cy² + Dx + Ey + F = 0, and with B = 0 the type is determined by A and C: equal gives a circle, same sign and unequal gives an ellipse, opposite signs gives a hyperbola, and exactly one of them zero gives a parabola.

The centre-radius form follows from the distance formula: (x − h)² + (y − k)² = r² states that every point is r from the centre, so it is the Pythagorean theorem written as a locus.

Converting from expanded form requires completing the square in both variables, the same procedure as for quadratics but applied twice. A negative result on the right indicates no real graph, and zero indicates the degenerate case of a single point.

Each conic also has a focus-directrix definition: a parabola is the locus equidistant from a focus and a directrix, which is why parabolic reflectors focus parallel rays to a point.