9.7
Geometria das Hipérboles
Uma hipérbole consiste em todos os pontos nos quais a diferença absoluta das distâncias a dois pontos fixos, chamados focos, permanece constante. A eq…
Uma hipérbole consiste em duas curvas abertas chamadas ramificações. Se P é um ponto na curva, então as distâncias de P aos dois focos são medidas, e a diferença absoluta dessas distâncias é constante. Não importa onde P é escolhido em qualquer ramo, essa diferença permanece a mesma.
A equação padrão para uma hipérbole centrada na origem e abertura ao longo do eixo x é x ao quadrado sobre a ao quadrado menos y ao quadrado sobre b ao quadrado é igual a um.
Cada ramo de uma hipérbole se aproxima de duas linhas diagonais, chamadas assíntotas, que guiam a curva até o infinito.
Duas vezes a dá a distância entre os vértices ao longo do eixo transversal, enquanto duas vezes b define o comprimento do eixo conjugado.
As equações das assíntotas dependem de a e b, que determinam as inclinações das linhas diagonais do retângulo central. Usando a fórmula ponto-inclinação, as equações das linhas assíntotas podem então ser escritas.
Hipérboles também aparecem em instrumentos ópticos. Em astronomia, o telescópio Cassegrain usa um espelho primário parabólico e um espelho secundário hiperbólico. Os focos primários recebem raios paralelos, e o secundário, compartilhando um foco com o primário, os reflete em direção ao seu segundo foco através de um orifício no primário para formar uma imagem.
View the full transcript and gain access to JoVE Core videos
Q1: What defines a hyperbola and how do its two branches relate to the foci?
A hyperbola consists of two open curves called branches, defined by a constant property: for any point P on either branch, the absolute difference of distances from P to two fixed points called foci remains constant. This defining relationship holds regardless of which branch or location on the curve you choose, making it a fundamental characteristic of hyperbolic geometry.
Q2: What is the standard equation of a hyperbola centered at the origin?
The standard equation for a hyperbola centered at the origin and opening along the x-axis is x² / a² − y² / b² = 1. Here, a represents the distance from the center to each vertex along the transverse axis, while b influences the shape and asymptote slopes. This form enables direct analysis and plotting of hyperbolic curves.
Q3: How do asymptotes guide a hyperbola, and what determines their slopes?
Each branch of a hyperbola approaches two diagonal lines called asymptotes that guide the curve toward infinity without ever touching them. The slopes of these asymptotes depend on both parameters a and b, which define the dimensions of a central rectangle. The diagonals of this rectangle, with dimensions 2a by 2b, directly determine the asymptote equations.
Q4: What does eccentricity measure in a hyperbola?
Eccentricity quantifies how open or spread a hyperbola is, always exceeding one for hyperbolas. It is calculated using the relationship between c (distance to foci) and a (distance to vertices). Understanding eccentricity helps distinguish hyperbolas from other conic sections like those studied in eccentricity of an ellipse, which have eccentricity less than one.
Q5: How are hyperbolic mirrors used in optical instruments like telescopes?
In a Cassegrain telescope, a hyperbolic secondary mirror works with a parabolic primary mirror to focus light precisely. The primary mirror focuses incoming parallel rays, and the hyperbolic secondary, sharing one focus with the primary, reflects them toward its second focus through a hole in the primary to form a clear image.
Q6: What is the relationship between the transverse and conjugate axes in a hyperbola?
The transverse axis defines the distance between the two vertices, measured as 2a, while the conjugate axis has length 2b. These perpendicular axes intersect at the center and determine the hyperbola's orientation and shape. Together, they form a central rectangle whose diagonals become the asymptotes guiding each branch.
Q7: Where are the foci located on a hyperbola, and how do they relate to the vertices?
For a hyperbola centered at the origin with horizontal transverse axis, the foci are located at (±c, 0), where c is calculated from the relationship c² = a² + b². The foci lie beyond the vertices on the transverse axis, and their separation determines the constant difference in distances that defines every point on the hyperbola.