When trying to decide whether a general equation is either a parabola, hyperbola, circle, or ellipse, you need to look at the signs, coefficients and variables that are squared. If only one variable is squared it's a parabola for example x^2+5y+3x+2=0 or (x+6)^2=7y. If both variables are squared and ONE is negative, it's a hyperbola. For example, Y^2- (x+3)^2/4=1. When differentiating an ellipse from a circle look at the coefficients. If they are the same its a circle; if they are different its an ellipse. Notice how in (x+6) + (y-3)=9 there is a common coefficient (1) and no denominator to indicate different radii. Therefore, its a circle instead of (x+6) + (y-3)/9=1 which would be an ellipse.
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These proofs provide a deeper understanding of conics through a visual proof of their foci relationships. They show a growth from the basic comprehension of conics I had when entering the class to the elevated level of knowledge I've acquired during the course.
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