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Conic Sections

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 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

      θ1 < θ2 ≤ 90°

      Point

      θ1 =θ2

      Line

      0 θ2 < θ1

      Hyperbola

 

  • 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 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

      θ1 < θ2 ≤ 90°

      Point

      θ1 =θ2

      Line

      0 θ2 < θ1

      Hyperbola

  • 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.
    • The fixed distance is called the radius of the circle.
  • To find the equation of a circle, let us watch the following video

  • The equation of a circle with radius r units and centre at (h, k) is…

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