Conic Sections

Introduction:

The curves parabola, hyperbola, ellipse and circle 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, etc.

Section of a Cone:

Let l be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle . Suppose we rotate the line m around the line l in such a way that the angle a remains constant. Then the surface generated is a double-napped right circular hollow cone.

Here, point V is called the vertex,

Line l is the axis of the cone,

Rotating line m is called a generator of the cone.

Intersection of a plane with a cone is called a conic section. We obtain different kinds of conic sections depending on the position of the intersecting plane with respect to the cone and by the angle made by it with the vertical axis of the cone. Let be the angle made by the intersecting plane with the vertical axis of the cone.

When the plane cuts the nappe (other than the vertex) of the cone, we have the following situations:

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

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

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

When the plane cuts at the vertex of the cone, we have the following different cases:

(a) When, then the section is a point .

(b) When , the plane contains a generator of the cone and the section is a straight line. It is the degenerated case of a parabola.

Circle:

ü A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.

ü The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle.

ü Standard form of equation of a circle:

where, (h,k) is the centre of the circle and r is radius of circle.

ü Simplest form of equation of a circle:

where, (0,0) is the centre of the circle and r is the radius of the circle.

ü General form of equation of a circle:

where, (-g, -f) is the centre of the circle and is the radius of the circle.

Question 1: Find the equation of the circle with centre (5,-2) and radius 3.

Solution:

Here, h=5, k=-2 and r=3

The equation of the circle is

Parabola:

A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of the parabola and the fixed point F is called the focus. A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola.

Standard equation of a parabola:

Length of latus rectum = 4a

Question 2: Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum, .

Solution:

The given equation of parabola is which is of the form

4a=20

a=5

Therefore, coordinates of focus are (5,0)

Axis of parabola is y=0

Equation of directrix is x=-5

i.e., x+5=0

Length of latus rectum=4x5=20

Ellipse:

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.

ü The two fixed points and are called foci (plural of focus)

ü The mid point of the line joining is called the centre of ellipse.

ü The line segment through the two foci and terminating at the boundary of the ellipse on both sides is called major axis.

ü The line segment through the centre of the ellipse, perpendicular to the line segment joining the two foci and terminating at the boundary of the ellipse on both sides is called minor axis.

ü The ends of major-axis are called vertices of the vertices of the ellipse.

ü The length AB of the major axis is denoted by 2a and the length of the minor axis denoted by 2b.

ü Semi-major axis = a.

ü Semi-minor axis=b.

ü The distance between two foci and is 2c.

Relation between semi-major axis (a), semi-minor axis (b) and the distance between the centre of the ellipse and one of the two foci (c):

Standard equation of an ellipse:

Latus rectum of an ellipse:

Eccentricity:

Question 3: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.