Class 11th mathematics(conic sections)

-
Let us consider a fixed vertical line (l) and let (m) be another line intersecting line (l) at any fixed point V. Let α be the angle of intersection between line l and m. If line m is rotated around line l such that the angle of intersection (α) remains constant. Then, the resultant surface is a double-napped circular hollow cone. Where V is known as the vertex, l is the axis and m is known as the generator of the resultant cone. The vertex separating the cone into 2 different parts is known as the nappes. The section obtained by the intersection of a cone with the plane is known a the conic section. Based on the angle and position of the intersecting plane w.r.t. cone, different types of conic sections can be obtained.

Conditions for Ellipse, Circle, Hyperbola and Parabola

When a plane cuts (other than vertex) the nappe of the cone, we have following conditions:
  • When β=900, the resultant is a circle
  • When α<β<900, the resultant is an ellipse
  • When β=α, the resultant is a parabola
  • When 0≤β<α, the resultant is a hyperbola

Degenerated Sections

When a plane cuts the vertex of the cone, we get following cases:
  • The section is a point when, α<β≤900
  • Parabola degeneration: When β=α, the resultant section is a straight line.
  • Hyperbola degeneration: When 0≤β<α, the resultant section is the pair of intersecting straight lines.
A set of points equidistant from a fixed point (center) in a plane is known the circle. Let ‘r’ be the radius of the circle and O(h, k) be the coordinate of its center. Let P(a, b) be any point on the circle. Then, the equation of the circle is given by:
(a−h)2+(b−k)2=r−−−−−−−−−−−−−−−−−−√[distance formula]
If the equation of parabola has y2 then, the symmetrical axis is along the x-axis and if the equation of parabola has x2, then the symmetrical axis is along the y-axis.
When the symmetrical axis is along the x-axis the parabola opens to the
  • Right, if x has a +ve coefficient,
  • Left, if x has -ve coefficient.
When the symmetrical axis is along the y-axis the parabola opens
  • Upwards, if y has a +ve coefficient
  • Downwards, if y has -ve coefficient.

Comments

Post a Comment

Popular posts from this blog

Class 11th science chemistry (thermodynamics)

-
Image
Thermodynamics deals with the change of energy in physical and chemical processes. It allows us to study these changes quantitatively as well as to make effective predictions. Therefore, we divide the universe as surrounding and system. Physical or chemical processes lead to progression or immersion of heat part of it may be converted to work. According to the first law of thermodynamics , these quantities are related as ∆U = q + w where q is heat and w is work. Change in internal energy is a state function and depends on only the initial and final states. On the other hand, w and q are not state functions. The change in temperature due to the transfer of heat one system to another system can be measured. Heat evolved or absorbed is expressed as q = C∆T. In case of expansion of gas, work is measured as w = –p ex ∆V Enthalpy Change Enthalpy change is defined as the change in energy of a system at constant pressure. It is denoted as ΔH. Enthalpy is a state function. Enthalpy ch...
Read more

Class 11th science chemistry (Redox reaction)

-
Image
Redox reaction is the combination of both oxidation and reduction reactions. When redox reaction takes place, both oxidation and reduction reactions also occur simultaneously. What is a Redox Reaction? Redox is a term used for the oxidation-reduction reaction. Redox reaction is a chemical reaction where a change in the oxidation state of atoms occurs. It involves electron transfer i.e if one chemical species gains electrons, then another chemical species gives or loses electrons. The species from which the electron is lost is said to be oxidized whereas the species to which the electron is added is said to be reduced. Example zinc displaces copper in an aqueous solution called copper sulfate. Redox reactions can be classified into four different categories: Combination reaction Decomposition reaction Displacement reaction Disproportionation reaction Oxidizing And Reducing Agents The role of an oxidizing agent (Oxidant) in a chemical reaction is to gain electrons. ...
Read more