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## Chapter 7 Conic Sections-Parabola, Hyperbola And Ellipse

7.1 Introduction: A conic section is the intersection of a plane and a cone. Hyperbola, ellipse and parabola are together known as conic sections, or just conics. They are so called because they are formed by the intersection of a right circular cone and a plane.

The General Equation for a Conic Section:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 7.2 Types of conics: There are three types of conics, hyperbola, ellipse, and parabola. The circle can be considered as a fourth type or as a kind of ellipse.

i) The circle and the ellipse are generated when the intersection of cone and plane is a closed curve.

ii) The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone -- for a right cone as in the picture at the top of the page this means that the cutting plane is perpendicular to the symmetry axis of the cone.

iii) If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola.

iv) In the remaining cases, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves

The type of sections can be found by using the formula;

B2– 4AC

• If B2 – 4AC > 0, then, the curve is a circle, ellipse, point or no curve.
• If B2 – 4AC = 0, then, the curve is a parabola, two parallel lines, 1 line or no curve.
• If B2 – 4AC < 0, then the curve is a hyperbola or two intersecting lines.

Through change of coordinates these equations can be put in standard forms:

• Circle: x2 + y2 = r2
• Ellipse: x2/a2 + y2/b2 = 1 , x2/a2 y2/b2 = 1
• Parabola: y2 = 4ax , x2 = 4ay,
• Hyperbola: x2 /a2 ̲ y2/b2 = 1, x2/a2 ̲ y2/b2 = -1
• Rectangular Hyperbola: xy = c2

Such forms will be symmetrical about the x-axis and for the circle, ellipse and hyperbola symmetrical about the y-axis.
The rectangular hyperbola however is only symmetrical about the lines y = x and y = − x. Therefore its inverse function is exactly the same as its original function.

These standard forms can be written as parametric equations,

• Circle: (acosθ,asinθ),
• Ellipse: (acosθ, bsinθ),
• Parabola: (at2, 2at),
• Hyperbola: (asecθ, btanθ) or (± a cosh u, b sinh u)
• Rectangular Hyperbola:  (ct, c/t)

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