Conic Sections
Introduction to Conic Sections

Conic sections or conics are the curves obtained by intersecting a doublenapped rightcircular 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 doublenapped cone (other than the vertex) by different ways.

If θ_{1} is the angle between the axis and the generator and θ_{2} is the angle between the plane and the axis, then for different conditions of θ_{1} and θ_{2}, we get different conics. These are described in the table shown below.

Condition
Conic Formed
Figure
θ_{2} = 90° (The plane cuts only one nappe of the cone entirely)
Circle
θ_{1} < θ_{2} < 90° (The plane cuts only one nappe of the cone entirely)
Ellipse
θ_{1} = θ_{2} (The plane cuts only one nappe of the cone entirely)
Parabola
0 ≤ θ_{2} < θ_{1} (The plane cuts each nappe of the cone entirely)
Hyperbola



The conic sections obtained by cutting a plane with a doublenapped cone at its vertex are known as degenerated conic sections.

If θ_{1} is the angle between the axis and the generator and θ_{2} is the angle between the plane and the axis, then for different conditions of θ_{1} and θ_{2}, we get different conics. These are described in the table shown below.

Condition
Conic Formed
Figure
θ_{1} < θ_{2} ≤ 90°
Point
θ_{1} =θ_{2}
Line
0 ≤ θ_{2} < θ_{1}
Hyperbola



Conic sections or conics are the curves obtained by intersecting a doublenapped rightcircular 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 doublenapped cone (other than the vertex) by different ways.

If θ_{1} is the angle between the axis and the generator and θ_{2} is the angle between the plane and the axis, then for different conditions of θ_{1} and θ_{2}, we get different conics. These are described in the table shown below.

Condition
Conic Formed
Figure
θ_{2} = 90° (The plane cuts only one nappe of the cone entirely)
Circle
θ_{1} < θ_{2} < 90° (The plane cuts only one nappe of the cone entirely)
Ellipse
θ_{1} = θ_{2} (The plane cuts only one nappe of the cone entirely)
Parabola
0 ≤ θ_{2} < θ_{1} (The plane cuts each nappe of the cone entirely)
Hyperbola



The conic sections obtained by cutting a plane with a doublenapped cone at its vertex are known as degenerated conic sections.

If θ_{1} is the angle between the axis and the generator and θ_{2} is the angle between the plane and the axis, then for different conditions of θ_{1} and θ_{2}, we get different conics. These are described in the table shown below.

Condition
Conic Formed
Figure
θ_{1} < θ_{2} ≤ 90°
Point
θ_{1} =θ_{2}
Line
0 ≤ θ_{2} < θ_{1}
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 ${\left(xa\right)}^{2}+{\left(yb\right)}^{2}={r}^{2}$.

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 ${\left(xr\right)}^{\mathrm{2\dots}}$
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