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Conic Sections




What is conic sections

  • The term 'conic sections' refers to the curves obtained by cutting a double cone with a plane. The four primary types of conic sections are circles, ellipses, parabolas, and hyperbolas.
  • Lets see how we obtained this conic sections
  • Consider a double cone, which is essentially two cones with the same base and axis but extending in opposite directions from the vertex.
  • Let $\beta$ be the angle made by the intersecting plane with the vertical axis of the cone and $\alpha$ is the slant angle of the cone
By Egm6322.s09.lapetina - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=6452778

Formation of a Circle

  • When a plane cuts the cone perpendicular to the axis of the cone and the cut is made in such a manner that it intersects one of the conical surfaces, the resulting curve is a circle. This means that the angle between the intersecting plane and the vertical axis of the cone is 90 degrees i.e $\beta=90$.
  • This is the case C
  • This formation aligns with the standard definition of a circle: a circle is the set of all points in a plane that are equidistant from a fixed point called the center. The center of the circle, in this case, would be the point where the plane intersects the axis of the cone, and the distance from this point to the curve (the radius) would be the distance from the center to any point on the conical surface within the confines of the intersecting plane.

Formation Of ellipse

  • When a plane cuts the cone at an angle to the axis of the cone and the cut is made in such a manner that it intersects one of the conical surfaces, the resulting curve is a ellipse. This means that the angle between the intersecting plane and the vertical axis of the cone is less than 90 degrees i.e $\alpha < \beta < 90$.
  • This is the case C

Formation of a Parabola:

  • A parabola is formed when the intersecting plane is parallel to the slant side (generatrix) of the cone but does not cut through the base of the cone. The curve produced by this intersection is a parabola. This implies that the plane is not perpendicular to the axis nor parallel to the base; instead, it's parallel to one of the sides of the cone. $\alpha < \beta < 90$
  • This is the case A

Formation of a Hyperbola:

  • A hyperbola is formed when a plane intersects both halves of the double cone . The curve produced by this intersection consists of two separate and open branches, each being a hyperbola.$0 < \beta < \alpha$
  • This is the case B


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