Have you ever struggled to learn conic sections. Well, starting with parabolas, as a math teacher whenever I used to introduce this concept to my students in high school, I was greeted with blank faces . It took a couple of classes with them to drive home the concept of parabola , its types and tangents to a parabola.
This concept is used in Calculus , in applications of derivatives and also in double and triple integrals, where we need to draw a rough sketch of the region of integration. I have written this post hoping to help you understand the topic of parabolas. All my students have developed a liking for mathematics and some of them have even chosen to continue their studies in Mathematics. So, I hope this post will be of help to you.
I have divided this post into 3 parts. The first part teaches you how a parabola is derived from a conic. It starts with a point on a parabola and shows how the ratio of the distance of the point from the focus = distance of the point from the directrix. It also teaches you how to find the equation of the axis. You can watch this video to understand this topic
The second part teaches you about the types of parabolas. There are 4 types of parabolas. It teaches you how to learn them through their diagrams. It shows you how to identify the focus, axis, latus rectum and directrix and how to find each of these. These parabolas have their vertices at the origin. You will also learn how to find all the above parameters when the vertex is not at the origin. You can access this topic by watching the video below.
I conclude with the equation of a tangent and condition of tangency to a parabola. There is a condition of tangency which you can use or if you find it difficult to remember, you can always substitute in the equation to calculate. A few problems are illustrated in the following video.
I hope this post was useful to you. Feel free to send me your feedback.