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Introduction
Conic sections play a crucial role in the study of geometry. According to how the plane intersects the right circular cone, different types of conic sections can be identified in mathematics.
Conic Sections
The term “conic” refers to a curve formed when a right circular cone is intersected by a plane. In Euclidean geometry, it has unique properties. The cone’s vertex divides it into two nappes, the upper and lower nappes, respectively.
A conic section is the result of a cone being intersected by a plane. Different types of conic sections are formed depending on the position of the plane that intersects the cone and the angle of intersection. Namely:
- Circle
- Ellipse
- Hyperbola
- Parabola
There are many examples of curves in everyday objects, such as the round silver mirrors you see at a metro station. Everything from planetary motion studies to the design of telescopes and satellites can benefit from curvature. A conic curve is formed by the intersection of a plane with a right circular cone that has two snaps. Let’s take a closer look at the cone’s various sections, their formulas, and their importance.
Conic Sections formula
Focus, eccentricity and Directrix
In other words, the locus of a point P moving in the plane of a fixed point F known as the focus (F) and of a fixed line d known as the directrix (with the focus off d) can be described as a conic section, and it is this locus that results in a constant e known as eccentricity for the distance of point P from focus F to the distance of d. Now,
- If eccentricity, e = 0, the conic is a circle.
- If 0<e<1, the conic is an ellipse.
- If e=1, the conic is a parabola.
- And, if e>1, it is a hyperbola.
Because the ellipse deviates from circularity, eccentricity measures the deviation. Suppose, the angle formed between the surface of the cone and its axis is β and the angle formed between the cutting plane and the axis is α, the eccentricity is;
e = cos α/cos β
Parameters of Conic
Conic sections define a few additional parameters in addition to focus, eccentricity, and directrix.
- Principal Axis: Ellipse or hyperbola has two focal points or foci that are connected by this line. The midpoint of the curve is where it all begins.
- Linear Eccentricity: The distance between a section’s focal point and the section’s centre.
- Latus Rectum: A sectional chord parallel to the directrix that runs through a focus.
- Focal Parameter: Distance between the focal point and the corresponding directrix.
- Major axis: The chord that connects the two points. The ellipse’s longest chord.
- Minor axis: An ellipse’s shortest chord.
Conic Sections standard forms
Circle – conic sections
Circular ellipse is an ellipse in which the cutting plane is perpendicular to the base. The centre of the circle serves as the focal point of the circle. The radius of a circle refers to the fixed distance the points on the circle’s locus have from the circle’s focus or centre. For a circle, the value of eccentricity (e) is e = 0. A circle has no axis of rotation. For a circle with a centre at (h,k) and radius r , the following is the general form of the equation:
(x−h)² + (y−k)² = r²
Conclusion
The intersection of a plane and a cone produces conic sections or sections of a cone. The parabola, hyperbola, and ellipse are the three primary sections of a cone (the circle is a special kind of ellipse). A cone’s various sections have various shapes, but they all share certain characteristics.
The plane intersects a double right circular cone in a conic section. We can create a variety of conics by varying the angle and location of the intersection. Circles, ellipses, hyperbolas, and parabolas are the four most common shapes. Because the cone has no vertices, no intersections can pass through it.
A cone’s various sections have various shapes, but they all share certain characteristics
