A conic section is a topic that has great significance in JEE main exam. Students can easily score from this topic if learned properly. 2-3 questions can be expected from this topic for any entrance exam. A conic section is the locus of a point moving in a plane such that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed line. The constant ratio is known as the eccentricity of the conic. The eccentricity shows how un-circular a curve is. Larger the eccentricity, lower curved it is Hyperbola.
Conic sections are obtained as intersections of a plane with a double-napped right circular cone.
The mathematician named Menaechmus introduced the conic sections. The applications of conics include the design of telescopes, automobile headlights, satellite dishes, etc. Hyperbola is a type of conic section in which two curves are like infinite bows. We can define it as an open curve having two branches that are mirror images of each other. The important terms related to the conic section are listed below.
- Axis of conic
- Vertex
- Chord
- Double ordinate
- Latus Rectum
Standard Equation
The standard equation is given by (x2/a2) – (y2/b2) = 1, where b2 = a2 (e2 – 1). Here e denotes the eccentricity.
Important Terms
(1) Auxilary Circle: It is a circle drawn with the endpoints of the transverse axis of the hyperbola as its diameter. The equation of the auxiliary circle is given by x2 + y2 = a2.
(2) Asymptotes: These are pairs of straight lines drawn parallel to the hyperbolas and assumed to touch them at infinity. The equations of the asymptotes are y = bx/a, and y = -bx/a respectively.
(3) Parametric Coordinates: We can denote the points on the hyperbolas using the parametric coordinates (x, y) = (a sec θ, b tan θ). These parametric coordinates denoting the points on the hyperbolas satisfy the equation (x2/a2) – (y2/b2) = 1.
Applications
Satellite systems make use of hyperbolic functions. When a satellite is launched into space, scientists use mathematical equations to predict the path. Astronomers also use hyperbolic functions to predict the path of the satellite to make adjustments so that the satellite gets to its final position. Televisions, microscopes, and telescopes also make use of the concept of hyperbolas.
Parabola
It is defined as 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. If the directrix is parallel to the y-axis, the standard equation of a parabola is given as y2 = 4ax. If the directrix is parallel to the x-axis, the standard equation is x2 = 4ay. For y2 = 4ax, the directrix is given by x = -a. Focus is S: (a,0). The length of the latus rectum is 4a. Vertex is (0,0). We can call parabolas an integral part of the conic section.
Students are advised to practise previous years’ question papers on conic section. Revising and solving chapter-wise questions will help students to understand the difficulty level of each chapter. Students can also download PDFs of important formulas of the conic section and JEE notes of conic section. Visit BYJU’S to download these for free.
