Create a blog outline about Unit of Circles. Indicate your references.

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Introduction to Conic Sections Conic sections are the figures generated when a plane intersects a double-napped cone. Depending on the position of the plane relative to the cone's axis, different curves are generated. The figure on the right shows what type of curve is created with each intersection between the cone and the plane. There are four types of conic sections: Circle, Parabola, Ellipse, and Hyperbola. When the plane intersects the vertex of the cone, a degenerate conic is created. The Four Types of Conic Sections Browse through the gallery below to see what the four conic sections are and their definitions! (Photo Source: GeoGebra) Parabola Circle Ellipse Hyperbola All conic sections can be expressed into the general equation: Ax? + Bxy + Cy? + Dx + Ey + F = 0, where A, B, C, D, E, and F are constants. We determine the type of conic sections as follows: If A = C, then the equation represents a circle. . If either A or C is zero, then the equation represents a parabola. If A and C have the same sign but not equal, then the equation represents an ellipse. If A and C have opposite signs, then the equation represents a hyperbola. The general equation can also be converted into standard equations. Watch this video by Math Guide to know more about this conversion:How do we apply conic sections in real life? Conic sections have applications in various fields: engineering, computer science, medicine, astronomy, and others! On the left are some of the everyday objects that use conic sections. Circles, for instance, may be used in trilateration, a method employed by GPS systems ("How GPS Receivers Work"). Trajectory of objects being thrown follow a parabolic path, and the equation of a parabola is very helpful in calculating several values related to the object's motion. Our planet travels in an elliptical orbit, and one has to know what an ellipse is in order to make sense of this phenomenon. Finally, evaporative cooling towers could not operate more efficiently if not for its hyperboloid shapes ("Why Do Cooling Towers"). These are just some of the applications of conic sections, and if you look closely around you, you might see a lot more! Conic Section Word Problem Let us try to answer a word problem to further appreciate the application of conic sections in real life situations. Here, we have a problem involving a parabola: PROBLEM: An engineer designs a satellite dish with a parabolic cross section. The dish is 8m wide at the opening, and the focus is placed 4m from the vertex. (a) Position a coordinate system with the vertex at the origin and the y-axis on the parabola's axis of symmetry and find an equation of the parabola. (b) Find the depth of the satellite dish at the vertex.SOLUTION: (a) Let us graph the parabola in a cartesian plane. As instructed, let our vertex be at (0,0) and our focus, which is 4 meters away from the vertex, be at (0,4). Let AB be the width of the parabola's opening and CD the depth of the satellite dish. F(0,4) C B miters (b) We want to find the dept of the satellite dish, that is we are looking for segment CD. First, let us find the equation of the parabola. Since the parabola has vertex at (0,0) and opens up, it must follow the equation, x2=4py, where p is the distance between the vertex and the focus. Since the focus is 4 meters away from the vertex, then the equation of our parabola is x2 = 16y. Now we need to find the y-coordinate of point C to know the depth of the satellite. The y-coordinate of point C is the same with the y-coordinate of point B. Since point B lies on the parabola, we can find its y-coordinate using the equation of the parabola which we generated earlier. However, we first need to know the x-coordinate of B. Since AB is 8 meters long and point C is there midpoint, CB is 4 meters long. Therefore, B has coordinates (4,y). To find y, we use substitute 4 to x in x2=16y. This gives us 42=16y -> 16=16y -> y=1. Therefore, the depth of the satellite is 1 meter. Finding Mathematics Everywhere Hopefully, the previous discussions make us even more aware about the applications of conic sections in things around us. They may not be immediately perceivable at first, but as soon as you're familiar with the concepts and ideas behind each conic section, it would be easy to connect them to our experiences. This is why consistent practicing is very important in order to go deeper into things and appreciate the value of mathematics