M4. Conic Section:

Conic sections are the geometric curve formed when a cone is cutting by a plane. 

When double-napped cone is slicing by a plane, then different types of conic sections are obtained as shown in figure. (Click here for more detail.)

The Conic section are divided into 4 - major categories as follows:

1) Circle

If a plane intersects a cone perpendicular to the axis, then the section is a circle, as shown in figure above.

2) Ellipse

If a plane intersects a cone at a given angle with the axis greater than the semi - vertical angle. then the section is an ellipse.

3) Parabola

If an intersection plane, not passing through vertex, is parallel to the generator of the cone, then the section is a parabola.

4) Hyperbola

If a plane intersects the double right cone such that the angle between the axis and the plane be less than the semi-vertical angle, then the section is a hyperbola.

» Generally, Conic Section is defined as:

The locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point to its distance from a fixed straight line is constant is called a conic section.

Where, the straight line is called directrix. The fixed point is called the focus. The constant ratio is an eccentricity (denoted by $e$). 

$eccentricity\;(e) = \frac{distance\;from\;focus\;to\;conic}{distance\;from\;conic\;to\;directrix}$

The straight line passing through the focus and perpendicular to the directrix is called the axis. The intersection of the curve and the axis is called the vertex.

A Conic section are classified by its eccentricity as follows:

If eccentricity $(e) = 0$; the conic is a Circle. 

If eccentricity $(e) = 1$; the conic is a Parabola. 

If eccentricity $(e) < 1$; the conic is an Ellipse. 

If eccentricity $(e) > 1$; the conic is a Hyperbola.

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1. Parabola:

Parabola has vertical axis

Let $p = (x,y)$ be a point on a parabola. The standard equation of a parabola with its vertex $(h,k)$ and focus is at $(h+a,k)$ is given by:

$(y-k)^2 = 4a(x-h)$; where $a≠0$

If a parabola is centered at the origin $(h,k)=(0,0)$ and focus is at $(a,0)$, then the equation of the parabola is:

$y^2 = 4ax$ ............ (i)

                                                                         Where $a≠0$; Parabola has a horizontal axis.

                                                                            Similarly,

                                                                           $x^2 = -4ay$ .............. (ii)

                                                                           Where $a≠0$; Parabola has a vertical axis.

» The important results of a parabola are tabulated below:

Equation of Parabola	Vertex	Focus	Equation of directrix	Axis	Length of latus Rectum
$y^2 = 4ax$	$(0,0)$	$(a,0)$	$x=a$	$y=0$	$4a$
$x^2 = -4ay$	$(0,0)$	$(0,a)$	$y=-a$	$x=0$	$4a$
$(y-k)^2 = 4a(x-h)$	$(h,k)$	$(h+a,k)$	$x= h-a$	$y=k$ parallel to $x-axis$	$4a$
$(x-h)^2 = 4a(y-k)$	$(h,k)$	$(h,k+a)$	$y=k-a$	$x=h$ parallel to $y-axis$	$4a$

2. Ellipse:

Ellipse has major axis $'a'$ as horizontal

Let an ellipse in the rectangular coordinate plane with the center at the origin $(0,0)$ and the major axis along the x-axis as shown in figure.

$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ .......... (i)

Where $'a'$ is major axis as horizontal and $'b'$ is minor axis as vertical.

Similarly,

                 $\frac{x^2}{b^2}+\frac{y^2}{a^2} = 1$ .......... (ii)

Where $'b'$ is a major axis as vertical and $'a'$ is minor axis as horizontal.

If center are not at origin, that is the center will be shifted to the point $(h,k)$ then the equation of ellipse becomes;

                 $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .......... (iii)

Where $'a'$ is major axis as horizontal and $'b'$ is minor axis as vertical.

Similarly,

                 $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ .......... (iv)

Where $'b'$ is a major axis as vertical and $'a'$ is minor axis as horizontal.

» The important results of a Ellipse are tabulated below:

3. Hyperbola:

Hyperbola opens right and opens left

The standard form of an equation of a hyperbola with a horizontal transverse axis and center at the origin $(0,0)$ is given by: 

$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$ .......... (i)

Where $a≠0$ and $b≠0$.

The $x$-term is positive. The branches of the hyperbola opens right and opens left.

The standard equation of an equation of a hyperbola with a vertical transverse axis and center at the origin $(0,0)$ is given by:

                $\frac{y^2}{a^2}-\frac{x^2}{b^2} = 1$ .......... (ii)

Where $a≠0$ and $b≠0$.

The $y$-term is positive. The branches of the hyperbola opens up and opens down.

» The important results of a Hyperbola are tabulated below:

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