Chapter 11

Conic Sections

Sections of a cone: circle, parabola, hyperbola, ellipse:    When the plane cuts the nappe (other than the vertex) of the cone, we have the following situations:

(a) When b = 90o, the section is a circle .

(b) When a < b < 90o, the section is an ellipse .

(c) When b = a; the section is a parabola .

(In each of the above three situations, the plane cuts entirely across one nappe of the cone).

(d) When 0 £ b < a; the plane cuts through both the nappes and the curves of intersection is a hyperbola .

Circle: Set of points in a plane equidistant from a fixed point. A circle with radius r and centre (h, k) can be represented as (x – h)+ (y – k)= r2

Parabola: Set of points in a plane that are equidistant from a fixed-line and point. A parabola with a > 0, focus at (a, 0), and directrix x = – a can be represented as y= 4ax

In parabola y= 4ax, the length of the latus rectum is given by 4a.

Ellipse: The sum of distances of a set of points in a plane from two fixed points is constant. An ellipse with foci on the x-axis can be represented as:  Hyperbola: The difference of distances of set of points in a plane from two fixed points is constant. The hyperbola with foci on the x-axis can be represented as:  