Families of Curves

作成者:Rafael Losada Liste

This activity belongs to the GeoGebra book GeoGebra Principia. In conclusion, the field to explore can expand indefinitely. As final examples with distances, let's observe some results involving powers. It is easy to demonstrate that the representation of XA2 + XB2 = k, with k constant, is a circle centered at the midpoint of A and B.

Note: The radius of that circle is sqrt(k/2 -((x(A)+x(B))²+(y(A)+y(B))²)/4).

From this, we deduce that the locus where the sum of the squares of distances to several points is constant is a circle centered at the midpoint of those points. Furthermore, taking D = XA2, we can observe that the real-plane representation of any polynomial p(D) is exclusively composed of one or more circles.

Note: This follows from the Fundamental Theorem of Algebra, since p(D) can be decomposed by factors (D − c), where c is a complex number. If c is non-negative real, then D − c = 0 corresponds to a circle with radius the square root of c. Otherwise, nothing is displayed.

Here, we also see that we can represent multiple curves of the same family, such as XAⁿ = XB and observe their behavior simultaneously.

Author of the construction of GeoGebra: Rafael Losada.

Families of Curves

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