CONIC SECTIONS
WHAT IS CONIC SECTION?
A conic section (or simply conic) is a curve obtained as the
intersection of the surface of a cone with a plane.
When the plane does pass through the vertex, the resulting figure
is a degenerate conic.
Depending on the angle of the plane with respect to the cone, a
conic section may be a circle, an ellipse, a parabola, or a hyperbola.
THERE ARE FOUR BASIC TYPES OF CONIC:
• If the right circular cone is cut by a plane perpendicular to
the axis of the cone, the intersection is a circle.
• If the plane intersects one of the pieces of the cone and
its axis but is not perpendicular to the axis, the
intersection will be an ellipse.
• To generate a parabola, the intersecting plane must be
parallel to one side of the cone and it should intersect one
piece of the double cone.
• Hyperbola, the plane intersects both pieces of the cone.
The general equation for any conic
section is
𝐴𝑥 2 + Bxy + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
where A, B, C, D, E and F are
constants.
• As we change the values of some of the constants, the shape of the
corresponding conic will also change. It is important to know the differences in
the equations to help quickly identify the type of conic that is represented by a
given equation.
If B 2 − 4 A C is less than zero, if a conic exists, it will be either a circle or an
ellipse.
If B 2 − 4 A C equals zero, if a conic exists, it will be a parabola.
If B 2 − 4 A C is greater than zero, if a conic exists, it will be a hyperbola.
STANDARD FORMS OF EQUATIONS OF CONIC SECTIONS:
• Circle
( x − h ) ² + ( y − k )² = r²
Center is ( h , k ) .
Radius is r .
• Ellipse with horizontal major axis
( x − h )² a ² + ( y − k ) ² b ² = 1
Center is ( h , k ) .
Length of major axis is 2 a .
Length of minor axis is 2 b .
Distance between center and either focus is c with c 2 = a 2 − b 2 , a > b > 0 .
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