Conic sections and circles

Take two ice-cream cones placed tip to tip: a double cone. It has a vertex (the shared tip), an axis (the line through its middle) and a semi-vertical angle α between the axis and the side.

Slice it with a flat plane that does not pass through the vertex. If the plane makes angle β with the axis, the cut edge is a curve called a conic section:

If the plane does go through the vertex you get a 'squashed' (degenerate) conic: a single point, one line, or a pair of lines.

The circle is the set of all points at a fixed distance r (the radius) from a fixed point (h, k) (the centre). By the distance formula:

• (x – h)² + (y – k)² = r².
• Centre (1, –3), radius 5: (x – 1)² + (y + 3)² = 25, which opens out to x² + y² – 2x + 6y – 15 = 0.
• Centre at the origin: simply x² + y² = r².

Going back: centre and radius from an equation. Group the x terms and the y terms and complete the square (add half the coefficient, squared, to both sides).

• x² + y² + 6x – 4y – 12 = 0 becomes (x² + 6x + 9) + (y² – 4y + 4) = 12 + 9 + 4.
• So (x + 3)² + (y – 2)² = 25: centre (–3, 2), radius 5.
• If x² and y² have a number in front, divide by it first: 3x² + 3y² – 6x + 12y = 0 becomes x² + y² – 2x + 4y = 0, so (x – 1)² + (y + 2)² = 5: centre (1, –2), radius √5.

A circle through given points. The centre is equally far from every point on the circle. Through (0, 0), (4, 0) and (0, 6):

• Equal distances to (0, 0) and (4, 0) put the centre on x = 2; to (0, 0) and (0, 6), on y = 3.
• Centre (2, 3), and r² = 2² + 3² = 13: (x – 2)² + (y – 3)² = 13.

Inside or outside? Compare a point's distance from the centre with r. For x² + y² = 25: (3, 4) gives 25, on the circle; (1, 4) gives 17, inside; (4, 4) gives 32, outside.