Conic Section

Conic Section

औपचारिक रूपमा, ग्रीकमा लगभग 500 to 200 BC को अवधिमा Conic sections पत्ता लगाइएको मानिन्छ। त्यतिखेर अपोलोनिस (200BC)ले Conic sections को बारेमा चर्चा गरेको पाईन्छ। यद्पि, सत्रौं शताब्दीको सुरुबाट मात्र Conic sections को व्यापक प्रयोग भएको पाईन्छ। आजका दिनहरूमा, प्रकृतिमा हुने धेरै प्रक्रियाहरूलाई मोडेल गर्न conic sections महत्त्वपूर्ण छन्। उदाहरण को लागी, universal bodies को locus कोनिक्स (सर्कल, अण्डाकार, प्याराबोला, र हाइपरबोला) हो ।

The term “conic section” refers to the geometric shapes formed by the intersection of a plane with a cone. The cone is not necessarily a right circular cone. There are many conic sections on the basis of the angle of cutting.  In every cases, conic section is the section of cone by a plane.

The ancient Greek mathematician Apollonius, also known as “Great Geometer,” made significant contributions to the understanding of conic sections around 200 BC. Apollonius explored the properties of ellipses, parabolas, and hyperbolas, categorizing them based on their unique characteristics.

Cone

In mathematics, cone is defined a three-dimensional surface
traced out by a straight line
passing through a fixed point and
moving around a fixed line.
In this definition of cone,
The straight line is called generator
The fixed point is called vertex
The fixed line is called axis

Conic भनेको डबल शंकुको cone लाई एउटा सतहले काट्दा बन्ने वक्र रेखा हो। सामान्यतया यसलाई right circular cone मा हेर्ने गरिन्छ, तर यो जुनसुकै cone मा पनि परिभाषित हुन्छ ।

Conic Section: The Geometry

Cone लाई एउटा plane ले काट्दा बन्ने plane curve (cross section) लाई conic section भनिन्छ ।

Conic section is a plane curve obtained by section (intersection) of a cone by a plane.
Based on this intersection, there are seven types of conic section.
These seven types of conic section are given below

According to this table, for example,
Parabola: a parabola is conic section defined as a plane curve obtained by intersection of a cone and a plane in which

• the plane misses the vertex and
• the plane is parallel to the generator.

Hyperbola: Hyperbole is a conic section obtained by section of a cone by a plane in which

• the plane misses the vertex of cone
• the plane is not parallel to the generator of cone

Similarly, we can define other conic sections

Notice that, from the table above we see that in some intersection, plane does not pass through the vertex of the cone. When the plane does pass through the vertex, the resulting conic is a degenerate conic section.

Conic Section: The Algebra

In analytic geometry, conic section can be defined in algebraic expression. This algebraic forms of conic section is called analytic representation.

In analytic geometry,
Conic section can be defined based on the definition of circle.

Circle

Circle is defined a locus of point whose
distance from a fixed point = constant

In this definition of circle,

• the constant distance is called radius.
• the fixed point is called center

Based on this definition of circle, we can define conic section.

Conic Section

Conic section is defined a locus of a point whose
\( \frac{\text{distance from a fixed point}}{\text{distance from a fixed line}} \)= constant
In this definition of conic section,

• the constant ratio is called eccentricity, it is denoted by e.
• the fixed point is called focus.
• the fixed line is called directrix.

Classification of Conic Section

Based on the value of e, conic section can be classified into three standard types. These three standard types are

1. Parabola (e =1)
2. Ellipse (e <1)
3. Hyperbola (e >1)

1. Circle (e =0)
2. Straight Line (e =∞)

General equation of Conic

If the general equation of second degree \( ax^2 +2hxy + by^2 +2gx + 2fy + c = 0\) represents a pair of striaght lines then the discriminat must be perfect square, thus
\((2hy+2g)^2-4(a)(by^2+2fy+c)\) is perfect square
or \((hy+g)^2-a(by^2+2fy+c)\) is perfect square
or \((h^2-ab)y^2+2(hg-af)y +(g^2-ac)\) is perfect square
Again, \((h^2-ab)y^2+2(hg-af)y +(g^2-ac)\) is perfect square if its discriminant is zero
Thus,
\( 4(hg-af)^2-4(h^2-ab)(g^2-ac)=0\)
or \( (hg-af)^2-(h^2-ab)(g^2-ac)=0\)
or \( abc+2fgh-af^2-bg^2-ch^2=0\)

Therefore, the discriminant is
\( \Delta= abc+2fgh-af^2-bg^2-ch^2 \)

1. \( Δ = 0, h^2=ab\) then the conic is pair of straight lines
2. \( Δ = 0, \frac{a}{h}= \frac{h}{b}=\frac{g}{f} \) then the conic is parallel lines
3. \( Δ \ne 0, a=b\ne 0, h=0\) then the conic is circle
4. \( Δ \ne 0, h^2 < ab\) then the conic is ellipse
5. \( Δ \ne 0, h^2=ab\) then the conic is parabola
6. \( \Delta \ne 0, h^2 >ab\) then the conic is hyperbola
7. \( Δ \ne 0, h^2 >ab, a+b=0\) then the conic is rectangular hyperbola

The equation of the conic whose center is at the origin is of the form
\( ax^2+by^2+2hxy+1=0 \)
This conic is called central conic

Parabola

Conic section is a plane curve obtained by section (intersection) of a cone by a plane.
So,
Parabola caan be defined as follows.

1. Parabola is conic section defined as a plane curve obtained by intersection of a cone and a plane in which
   the plane misses the vertex and
   the plane is parallel to the generator

2. Parabola is conic section defined as a plane curve obtained by intersection of a cone and a plane in which if
   α= angle between generator and axis and
   β= angle between plane and axis,
   and
   α=β
   then
   the section is a parabola,
   in which
   eccentricity = \(\frac{\cos \beta}{\cos \alpha}\)
   here
   the eccentricity is the measure of how far the conic deviates from being circular

3. Parabola is a plane curve defined a locus of a point in which
   \( \frac{\text{distance from a fixed point}}{\text{distance from a fixed line}} \)= constant (e=1)

   In this definition of conic section,
   the constant ratio is called eccentricity, it is denoted by e.
   the fixed point is called focus.
   the fixed line is called directrix.
   

4. Parabola is a plane curve defined a locus of a point in which the distance from a fixed point (or focus) and distance from a fixed line (or directrix) is always equal.
5. Parabola is a plane curve defined a locus of a point which is always equidistant from a fixed point (or focus) to a fixed line (or directrix)

• Focus: The fixed point of parabola is called focus
• Directrix: The fixed line of parabola is called directrix
• Axis: The straight line passing through focus and perpendicular to directrix is called axis
• Vertex: The meeting point of axis and parabola is called vertex
• Latus rectum: the chord passing through focus and perpendicular to axis is called latus rectum.
  The distance between the meeting points of latus rectum to the parabola is called length of latus rectum.

Equation of Parabola

Let C be a parabola whose
Focus is F (a,0)
Directrix is \(l: x = -a\)
Vertex is O: (0,0)
Take any point P(x,y) on parabola C,

1. Draw PA ⊥ \(l\) then A (-a,y)
2. Join F and P

By the definition of parabola
PA = PF
or \( (x+a)^2=(x-a)^2+y^2 \)
or \(x^2+2ax+a^2=x^2-2ax+a^2+y^2\)
or \(2ax=-2ax+y^2 \)
or \(y^2=4ax \)

Summary on Equation of Parabola

The basic parameters of parabola are summarized as below

Ellipse

Conic section is a plane curve obtained by section (intersection) of a cone by a plane.
So,
Ellipse can be defined as follows.

1. Ellipse is conic section defined as a plane curve obtained by intersection of a cone and a plane in which
   the plane misses the vertex and
   the plane is NOT at right angle with the axis

2. Ellipse is conic section defined as a plane curve obtained by intersection of a cone and a plane in which if
   α= angle between generator and axis and
   β= angle between plane and axis,
   and
   α < β < 90
   then
   the section is a ellipse,
   in which
   eccentricity = \(\frac{\cos \beta}{\cos \alpha}\)
   here
   the eccentricity is the measure of how far the conic deviates from being circular

3. Ellipse is a plane curve defined a locus of a point in which
   \( \frac{\text{distance from a fixed point}}{\text{distance from a fixed line}} \)= constant (e <1)

   In this definition of conic section,
   the constant ratio is called eccentricity, it is denoted by e.
   the fixed point is called focus.
   the fixed line is called directrix.
   

4. Ellipse is a plane curve defined a locus of a point in which the distance from a fixed point (or focus) is always less than the distance from a fixed line (or directrix)
5. Ellipse is a plane curve defined a locus of a point whose sum of distances from two fixed points (foci) is always a constant.

In this definition

• Foci: The two fixed points of ellipse
• Directrix: The two fixed line of ellipse
• Axis: The straight line passing through focus and perpendicular to directrix
• Major axis: The straight line passing through the foci
• Minor axis: The straight line passing through the center and perpendicular to the major axis
• Length of major axis: The distance between the vertices on major axis
• Length of minor axis: The distance between the meeting points of minor axis with ellipse
• Vertices: The meeting points of the major axis with ellipse
• Centre of ellipse: The middle point of the join of foci
• Latus rectum: The chord passing through focus and perpendicular to major axis

Please Note that

1. Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis.
2. Each endpoint of the major axis is the vertex of the ellipse, and each endpoint of the minor axis is a co-vertex of the ellipse.
3. The center of an ellipse is the midpoint of both the major and minor axes.
   The axes are perpendicular at the center.
4. The foci always lie on the major axis

There are four variations of the standard form of the ellipse. These variations are categorized first by
the location of the center (the origin or not the origin),
and by
the position (horizontal or vertical).

Ellipse: Proof of Basic Facts

1. c=ae

   If Z and Z’ are the directrix, F(c,0) and F'(-c,0) are the foci, and A(a,0) and A;(-a,0) are the vertices of ellipse, then by the definition of conic we have
   \(\frac{A’F}{A’Z}=e\)
   or \(A’F=A’Z e\) (1)
   Similarly, we have
   \(\frac{AF}{AZ}=e\)
   or \(AF=AZ e\) (2)
   Substracting (2) from (1), we get
   \(A’F-AF=(A’Z-AZ) e\)
   or\((A’O+OF)-(AO-OF)=(A’A) e\)
   or\(2OF=2a e\)
   or\(2c=2a e\)
   or\(c=a e\)
   

2. sum of distance from Foci is 2a.

   the distance of a point A(a,0) from fous F(c,0) is
   a−(c)=a-c.
   the distance of a point A(a,0) from fous F'(-c,0) is
   a−(-c)=a+c
   The sum of the distances from Foci is
   (a+c)+(a−c)=2a
   

3. Relation between a, b, c.

   \(2 \sqrt{b^2+c^2}=2a\)
   or\(b^2+c^2=a^2\)

Equation of Ellipse

Let C be an ellipse whose
foci are (−c,0) and (c,0).
center is O: (0,0)
Take any point P(x,y) on ellipse C,

Then,

If (a,0) is a vertex of the ellipse, then the distance from (−c,0) to (a,0) is
a−(−c)=a+c.
The distance from (c,0) to (a,0) is
a−c.
The sum of the distances from the foci to the vertex is
(a+c)+(a−c)=2a
If (x,y) is a point on the ellipse, then we can define the following variables:
d1=the distance from (−c,0)to (x,y)
d2=the distance from (c,0)to (x,y)
By the definition of an ellipse,
d1+d2=2a


Here
\(d_1+d_2=2a\)
or \(\sqrt{(x+c)^2+y^2}+\sqrt{(x−c)^2+y^2}=2a\)
or \(\sqrt{(x+c)^2+y^2}=2a-\sqrt{(x−c)^2+y^2}\)
or \((x+c)^2+y^2=\left [2a-\sqrt{(x−c)^2+y^2} \right] ^2\)
or \(x^2+2xc+c^2+y^2=4a^2-4a\sqrt{(x−c)^2+y^2} +(x−c)^2+y^2 \)
or \(x^2+2xc+c^2+y^2=4a^2-4a\sqrt{(x−c)^2+y^2} +x^2-2xc+c^2+y^2 \)
or \(2xc=4a^2-4a\sqrt{(x−c)^2+y^2} -2xc\)
or \(4xc-4a^2=-4a\sqrt{(x−c)^2+y^2}\)
or \(xc-a^2=-a\sqrt{(x−c)^2+y^2}\)
or \(\left[xc-a^2\right]^2= \left[ -a\sqrt{(x−c)^2+y^2} \right ]^2\)
or \(c^2x^2-2a^2cx+a^4= a^2 \left[(x−c)^2+y^2 \right ]\)
or \(c^2x^2-2a^2cx+a^4= a^2 (x^2-2xc+c^2+y^2 ) \)
or \(c^2x^2-2a^2cx+a^4= a^2x^2-a^22xc+a^2c^2+a^2y^2 \)
or \(a^2x^2-c^2x^2+a^2y^2=a^4-a^2c^2 \)
or \(x^2(a^2-c^2)+a^2y^2=a^2(a^2-c^2) \)
or \(x^2b^2+a^2y^2=a^2b^2 \) \(a^2-c^2=b^2\)
or \(\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \)

Thus, the standard equation of an ellipse is
\(\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \)
This equation defines an ellipse centered at the origin.


NOTE
1. If a>b, the ellipse is stretched further in the horizontal direction
2. if b>a the ellipse is stretched further in the vertical direction.
3. When c = 0, both foci merge together with the center of the ellipse and so the ellipse becomes a circle
4. When c = a, then b = 0. The ellipse reduces to the line segment joining the two foci

Summary of parameters in an Ellipse

	Ellipse	Ellipse	Ellipse	Ellipse
Equation	\( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\); a > b	\( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\); a < b	\( \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\); a > b	\( \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\); a < b
Center	(0,0)	(0,0)	(h,k)	(h,k)
Vertex	\( (\pm a,0) \)	\( (0,\pm b) \)	\( (h\pm a,k) \)	\( (h,k\pm b) \)
Focus	\( (\pm ae,0) \)	\( (0,\pm be) \)	\( (h\pm ae,k) \)	\( (h,k\pm be) \)
Directrix	\( x=\pm \frac{a}{e} \)	\( y=\pm \frac{b}{e} \)	\( x=h\pm \frac{a}{e} \)	\( y=k\pm \frac{b}{e} \)
Length Rectum	\( 2 \frac{b^2}{a} \)	\( 2 \frac{a^2}{b}\)	\( 2 \frac{b^2}{a} \)	\( 2 \frac{a^2}{b}\)
Eccentricity	\( b^2=a^2(1-e^2) \)	\( a^2=b^2(1-e^2) \)	\( b^2=a^2(1-e^2) \)	\( a^2=b^2(1-e^2) \)