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Name: Calculus Connections Proiect 4: Identifying Conic Sections Without Completing the Square So Far, we have seen many equations of conic sections on which we had to complete the square iorderjo find our critical information. Now that we have seen all three major conic sections parabolas, ellipses, and hyperbolas we will be presented sometimes with equations that not only need to have the square completed but may not state which conic section they are! Luckily, there is a way to tell. TH EDREM The standard equation if + Cl" +r+ ET +F = D represents the equation of a conic as-long- as A and C are not simultaneously zero {in this case; we would have a line}. Using this equation; we can determine the Following pieces ofinformation: AC I: D, the conic section is a hyperbole. AC: CI J the conic section is an ellipse. 2} If 3} If J the conic section is a parabola. Let's now consider the case 2, where do . This means that either A or C must be negative, but not both. If both A and C were negative, AC would be positive. Thus we have the following two possible equations: - Ax' + Cy + Dx+ Ey + F =0 and 4x - Cy' + Dx + Ev + F =0 Let's look at - Ax + Cy'+ Dx + Ey+ F =0 - Ax' + Cy' + Dx + Ey + F =0 is equivalent to Cy' + Ey - Ax' + Dx + F= or = -F Now we may complete the square to get into standard form.E D = - -F 2.A Factoring our binomials we get: 2 D 2 D 2 = -F 20 4C 2A 4A Combining like terms gives: 2 - Ax D =-F+ E D 20 2.A AC 4A In order to get a coefficient of 1 on the right hand side, we now can divide through. 2 E D X- 20 2.A E E - F + . D F 40 4C 4.A Although the denominator does look cumbersome, it is a constant and thus we do have the standard form of a hyperbola! The case of Ax - Cy + Dx + Ey + F =0 follows similarly. Now it's your turn! Using the same methodology, variables, and constants from above, prove that the AC rule is true for ellipses (case 1) and parabolas (case 3). Good luck and have fun @