Conic Section Secrets: Parabola, Ellipse, Hyperbola-Class 11
Excel in Conic Sections with my online math classes
Welcome to the world of Conics-Parabola, Ellipse and Hyperbola. The curves known as Conics were named after their discovery as the intersection of a plane with a right circular cone. Depending on the angle in which the plane intersects the cone, we get a circle, parabola, ellipse or hyperbola.
This topic requires a little understanding, so you can contact me for online tutoring if you need help. You can make use of the free youtube videos at the end of this page. You can choose to learn specific topics or the entire syllabus.
In Class 11, you'll be learning Parabola, Ellipse and Hyperbola. A parabola is the set of all points in a plane which are equidistant from a fixed point and a fixed line. The fixed point is called the focus and the fixed line is called a directrix.
Basically, there are four standard forms of a parabola, which you need to remember. As I tell my students, up, down, left and right. For each of these, you need to understand the equation of the axis, directrix, coordinates of the focus and lenght of the latus rectum.
Note that the latus rectum of a conic is a line segment passing through the focus and perpendicular to the axis. You'll need to draw these parabolas to get a clearer understanding. Another point to remember, is that, in all of these, the origin is the vertex.
Practice solving problems on these. Note that you'll use parabola and ellipse in Class 12 as well, so you need to understand it well. Also learn how to calculate the equation of a parabola and its other components if the vertex is shifted from the origin.
Moving on to ellipse, note that an ellipse is the set of points whose distance from a fixed point is equal to e times its distance from a fixed line. An ellipse has two fixed points, foci and two fixed lines, directrices. e stands for the eccentricity which is a real number lying between 0 and 1.
Note that for a parabola, e = 1. There are two basic equations of an ellipse and I highly recommend drawing it to get a clearer understanding. You'll learn to calculate the center, major and minor axes, their lengths, vertices, foci, length of the two latus rectum and the equations of the two directrices.
Again, practice in problem solving is essential and learn how to calculate each part of an ellipse if the origin is shifted. Also learn how to calculate each of the parameters if the center of the ellipse is shifted from (0,0). Coming to hyperbola, a hyperbola is the locus of a point which moves so that it's distance from a fixed point (focus) is equal to e times its distance from the fixed line (directrix)where e >1.
Properties of a hyperbola and ellipse are more or less similar. There are two types of hyperbola. You have the center, the transverse and conjugate axes, the length of these, the vertices, foci, length of the latus rectum and the equations of the directrices. A hyperbola in which a = b is called a rectangular hyperbola.
Again, as in the case of the other conics, you'll need to learn how to calculate the components of a hyperbola if the origin is shifted. Next, given a second degree equation in x and y, you'll learn how to identify if the given equation represents a parabola, ellipse, hyperbola, a circle or a pair of straight lines. Moving on, a very important concept is the condition of tangency.
Learn when the line y = mx + c, will touch a parabola, ellipse or a hyperbola. An interesting point to note is that a tangent at a point is perpendicular to the radius at that point. The radius can also be calculated using the formula for perpendicular from a point to a line.
To learn more about Conics, for Class 11, you can join my online classes. Learn how to answer conceptual based questions, get asessments every week and most important boost your scores.