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 ।
- Circle
- Ellipse
- Parabola
- Hyperbola
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
| Conic | Vertex | Generator | Axis | |
| 1 | Point | Cuts | outside | |
| 2 | Line | Cuts | touches | |
| 3 | Line Pair | Cuts | inside | |
| 4 | Circle | Misses | Right angle | |
| 5 | Ellipse | Misses | Not right angle | |
| 6 | Parabola | Misses | Parallel to | |
| 7 | Hyperbola | Misses | Not Parallel to |
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
- Parabola (e =1)
- Ellipse (e <1)
- Hyperbola (e >1)
- Circle (e =0)
- 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
\( \Delta= abc+2fgh-af^2-bg^2-ch^2 \)
- \( Δ = 0, h^2=ab\) then the conic is pair of straight lines
- \( Δ = 0, \frac{a}{h}= \frac{h}{b}=\frac{g}{f} \) then the conic is parallel lines
- \( Δ \ne 0, a=b\ne 0, h=0\) then the conic is circle
- \( Δ \ne 0, h^2 < ab\) then the conic is ellipse
- \( Δ \ne 0, h^2=ab\) then the conic is parabola
- \( \Delta \ne 0, h^2 >ab\) then the conic is hyperbola
- \( Δ \ne 0, h^2 >ab, a+b=0\) 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
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