Conic Sections: Circle and Parabola

Introduction to Conic Sections

• Conic sections or conics are the curves obtained by intersecting a double-napped right-circular cone with a plane.

  • The concept of conic sections is widely used in astronomy, projectile motion of an object, etc.

  • The example of conic sections are circle (Figure I), ellipse (Figure II), parabola (Figure III) and hyperbola (Figure IV).
    

• Different types of conics can be formed by intersecting a plane with a double-napped cone (other than the vertex) by different ways.

  • If θ₁ is the angle between the axis and the generator and θ₂ is the angle between the plane and the axis, then for different conditions of θ₁ and θ₂, we get different conics. These are described in the table shown below.

    Condition	Conic Formed	Figure
    θ2 = 90° (The plane cuts only one nappe of the cone entirely)	Circle
    θ1 < θ2 < 90° (The plane cuts only one nappe of the cone entirely)	Ellipse
    θ1 = θ2 (The plane cuts only one nappe of the cone entirely)	Parabola
    0 ≤ θ2 < θ1 (The plane cuts each nappe of the cone entirely)	Hyperbola

    
     

• The conic sections obtained by cutting a plane with a double-napped cone at its vertex are known as degenerated conic sections.

  • If θ₁ is the angle between the axis and the generator and θ₂ is the angle between the plane and …