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 the circle with radius r and centre (0, 0) is $x^2+y^2=r^2$.

• The equation of the circle with centre (a, b) and radius r is ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$.

• General equation of the circle is $x^2+y^2+2gx+2fy+c=0$, where $\left(-g, -f\right)$ is the centre and $r=\sqrt{g^2+f^2-c}$ is the radius of the circle.

• The equation of a circle with $\mathrm{A}\left(x_1, y_1\right)$ and $\mathrm{B}\left(x_2, y_2\right)$as the extremities of a diameter is $\left(x- x_1\right)\left(x-x_2\right) +\left(y- y_1\right)\left(y-y_2\right)=0$.
  
   

Equation of Circle in Different Conditions

      1. The equation of the circle with radius r, touching both the axes and lying in the first quadrant is ${\left(x-r\right)}^{\mathrm{2\dots }}$