Conic Tutorial

The sections of a cone are  ellipses, Circles, parabolas and hyperbolas. Because these diagrams can be obtained by passing a plain through vertex of the double-napped cone. In this conic tutorial we are going to see the sections of the cone.

Circle - conic tutorial:
The circle in the section of a cone is like ellipse. When two foci are coincided to each other the circle is formed. This conic section can be formed when a circular cone is intersected with a perpendicular plane.

Sections of a cone - Parabolas - conic tutorial:

One of the section of the cone is the parabola. The points will not be on the same line but in the plain are equal from a fixed line and in a fixed point.

Solve Maxima and Minima:
The local minimum and the local maximum value must be known for finding the maxima and minima values.

Ellipse - conic tutorial:
The section of a cone is an ellipse. It is from a constant two fixed point with the sum of its focus of all the points. The concept of an ellipse makes different terminologies like focus of ellipse, loci of ellipse, etc.

Equation of ellipse:
Equation of ellipse is `(x - h)^2 / a^2 ` + `(y - k)^2 / b^2` = 1 . here real numbers are h, k, a and b.

Sections of a cone-General Equation of an Ellipse

The points of locus will be F ( p, q ) and the equation of directrix ax + by + c = 0. This is the general equation of ellipse

Theorem on Line contact with an Ellipse:
The condition for the line y = m x + c to touch the ellipse `(x^2/a^2)+(y^2/b^2) ` = 1 is that c = ±  `sqrt(a^2 . m^2 + b^2)` .

Finding the point of contact for Ellipse: 
The tangent y = m x + `sqrt(a^2 . m^2 + h^2)`  touches the ellipse `(x^2/a^2)+(y^2/b^2) `= 1, at the point `[(-a^2m/sqrt(a^2m^2+b^2) , b^2/sqrt(a^2m^2+b^2)]`

Hyperbola - conic tutorial:

A hyperbola is the section of a cone. A conic which an eccentricity greater than 1. It has two branches extending up to infinity, two foci, two directions, two vertices and two axes. ( i . e . )  transverse and conjugate and has a center which is the midpoint of the line of two foci and perpendicular conjugate axis is the line through the center and to the transverse axis.