Conic sections can be obtained as intersections of a plane with a double-napped right circular cone. The applications of these curves include the design of telescopes, automobile headlights, and reflectors in flashlights, etc. The constant ratio is known as the eccentricity of the conic. The eccentricity of a circle is zero. It shows how “un-circular” a curve is. Larger the eccentricity, lower curved it is.
If we consider the interaction of a plane with a cone, the section so obtained is called a conic section. Hence, the curves obtained by intersecting a right circular cone by a plane are conic sections. As far as the JEE exam is concerned, this is an important topic. The important terms are listed below.
- Axis of conic
- Vertex
- Chord
- Double ordinate
- Latus Rectum
The line passing through focus, perpendicular to the directrix is the axis of conic. The point of intersection of conic and axis is the vertex. The line segment joining any 2 points on the conic is called the chord. Chord perpendicular to the axis is the double ordinate. Double ordinate passing through focus is the latus rectum.
Let β is the angle made by the intersecting plane with the vertical axis of the cone. When the plane cuts the nappe of the cone, we have the following situations.
1. If β = 900, the section will be a circle.
2. If α < β < 900, the section is an ellipse.
3. If 0 ≤ β < α, the curves of intersection is a hyperbola.
Parabola
A U-shaped plane curve where any point is at an equal distance from a fixed point (known as the focus) and from a fixed straight line, the directrix is called a parabola.
Standard Equation
The equation is y2 = x when the directrix is parallel to the y-axis. If the directrix is parallel to the y-axis, the standard equation is given as y2 = 4ax.
If the directrix is parallel to the x-axis, the standard equation is x2 = 4ay.
Standard Parabola
Standard equation: y2 = 4ax
Directrix : x = -a
Focus: S: (a,0)
Length of latus rectum: 4a
Vertex: (0,0)
Important results
1. 4 × distance between vertex and focus = Latus rectum = 4a.
2. 2 × Distance between directrix and focus = Latus rectum = 2(2a).
3. Point of intersection of Axis and directrix and the focus is bisected by the vertex.
4. The focus is the midpoint of the Latus rectum.
5. (Distance of any point from the axis)2 = (LR) (Distance of the same point from tangent at the vertex)
Parabola is symmetric with respect to the axis. If the equation has the term y2, then the axis of symmetry is along the x-axis. If the equation has the term x2, then the axis of symmetry is along the y-axis. When the axis of symmetry is along the x-axis, it opens to the right if the coefficient of x is positive and to the left, if the coefficient of x is negative.