CONIC SECTIONS

Conic Sections

These curves are 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 motions, design of telescopes and antennas, reflectors in flashlights and automobile headlights etc.

Sections 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 α.

Suppose we rotate the line m around the line l in such a way that the angle α remains constant. 

Then the surface generated is a double-napped right circular hollow cone herein after referred to as cone and extending indefinitely far in both directions.

• The point V is called the vertex; the line l is the axis of the cone.
•  The rotating line m is called the generator of the cone. 
• The vertex separates the cone into two parts called nappes.

If we take the intersection of a plane with a cone, the section so obtained is called a conic section. 

Thus conic sections are the curves obtained by intersecting a right circular cone by a plane.

When plane cuts the nappe of the cone we have following :

• When β = 90°, the section is a circle.
• When α < β < 90° , the section is an ellipse.
• When β = α; the section is a parabola.
• When 0 ≤ β < α; the plane cuts through both the nappes and the curves of intersection is a hyperbola.

Degenerated conic sections

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

• When α < β ≤ 90° , then the section is a point.
• When β  = α, the plain contains a generator of the cone and the section is a straight line. It is the degenerated case of a parabola.
• When 0 ≤ β < α, the section is a pair of intersecting lines. It is the degenerated case of a hyperbola.

Circle

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

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.

Given circle C( h, k ) be the centre and r the radius of circle. Let P( x, y ) be any point on the circle. Then equation of circle is given by the formula :

( x – h )² + ( y – k )² = r²

Parabola

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.

‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air.

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.

Equations of parabola

y² = 4ax,  y² = - 4ax, x² = 4ay, x² = - 4ay,

Letus rectum = 4a

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.

• The two fixed points are called foci ( plural of focus ) of the ellipse.
• The mid point of the line segment joining the foci is called the centre of the ellipse. 
• The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called minor axis.

The end points of the major axis are called the vertices of the ellipse

• the length of the major axis is 2a
• the length of the minor axis is 2b
• the distance between the foci is 2c

We have   a² = b² + c² ,

Ecentricity e = c/a

Equation of ellipse

x²/a² + y²/b² = 1  

Letus rectum, l = 2b²/a