In this Chapter, we will study about some curves - circles, ellipses, parabolas and hyperbolas. The names parabola and hyperbola are given by Apollonius. These curves are in fact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone.
These curves have a very wide range of applications in fields such as planetary motion, design of telescopes and antennas, reflectors in flashlights and automobile headlights.
A circle is the set of all points in a plane that are equidistant from a fixed point in the plane. The equation of a circle with centre (h, k) and the radius r is (x - h)2 + (y - k)2 = r2.
A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane. The equation of the parabola with focus at (a, 0) a > 0 and directrix x = -a is y2 = 4ax.
Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola. Length of the latus rectum of the parabola y2 = 4ax is 4a.
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse.
The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.
A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.
The eccentricity of a hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola.