Page 72 - Calculus for the Clueless, Calc II
P. 72
Chapter 12—
Conic Sections—Circle, Ellipse, Parabola, Hyperbola
Most books call the circle, parabola, ellipse, and hyperbola conic sections without explaining why. These curves
are found by passing a plane through the truncated (cut-off) right circular cone pictured here. They are formed
as follows:
Circle Plane parallel to the top or bottom
Ellipse Plane on the top or bottom not parallel to the top or bottom but hitting all parts of the outside.
Parabola A plane parallel to an edge of the cone
Hyperbola Figure formed by a plane intersecting the top and bottom
Definition
Circle—The set of points that are equidistant from a point, the center (h, k). That distance is r, the radius.
The distance formula:
2
2
2
Squaring, we get (x - h) + (y - k) = r .
Example 1—
2
2
Find the radius and center if (x - 4) + (y + 1/2) = 11. r = 11 1/2, center (4, - 1/2).
Find the center and radius of the circle 2x + 2y + 8x - 16y + 6 = 0. In order to do this, we have to complete the
2
2
square, something we have not done since the derivation of the quadratic formula.
