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.
We have four conic sections, each with their own distinct foci relationships. The locus of points to foci for a hyperbola all have the same difference between the two foci. What this means is that the difference between the distances of any point on the hyperbola and the two foci will be the same. For a parabola the distance between a point and the foci is the same as the distance between that point and the directrix. For an ellipse, the sum of the distances between any point and the foci are congruent. For a circle all points are equidistant to the center, the focus. This is basically just defining a radius, a familiar concept, but treating the center as a focus!
Click on the links provided for a visual display of each conic section and a proof of their relationship with their foci! |
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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