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Guilherme Mazui

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What is the Difference Between Hyperbola and Ellipse?

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Both the hyperbola and ellipse are types of conic sections, which are the intersection of a plane with a double cone. They have some similarities, such as being symmetrical about their main and minor axes, but they also have distinct differences. Here are the main differences between a hyperbola and an ellipse:

  • Position of the Directrix: The position of the directrix varies between the two shapes. In an ellipse, the directrix is outside the semi-major axis, while in a hyperbola, the directrix is inside the semi-major axis.
  • Standard Equations: The standard equations for hyperbolas and ellipses differ by the sign of the constant term. The standard equation for an ellipse is given by [Ax^2 + By^2 = C], where A and B are positive constants, while the standard equation for a hyperbola is given by [Ax^2 - By^2 = C], where A and B are positive constants.
  • Shape: The shape of an ellipse ranges between a straight line segment at one limit and a circle at the other limit. The shape of a hyperbola, on the other hand, ranges between a parabola at one limit and a straight line at the other limit. In contrast, parabolas range in shape between a straight line segment at one limit and a curve that curves outward forever at the other limit.
  • Curvature: The curvature of a hyperbola that is nearly parallel to the cone keeps curving outward forever, but with a little bit more curvature close to the vertex than in a parabola. A hyperbola that is nearly vertical has nearly all of its curvature close to the vertex and almost no curvature far away.
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  • What is the Difference Between Hyperbola and Ellipse?
  • Comparative Table: Hyperbola vs Ellipse

    • Comparative Table: Hyperbola vs Ellipse

    Here is a table summarizing the differences between a hyperbola and an ellipse:

    Feature Ellipse Hyperbola
    Shape Closed curve Open curve
    Perimeter Finite Infinite
    Center (0, 0) (0, 0)
    Vertex Yes Yes
    Focus Yes Yes
    Equation (Standard form) x^2/a^2 + y^2/b^2 = 1 x^2/a^2 - y^2/b^2 = 1
    Tangent Equation (Slope form) y = mx + √(a^2m^2 + b^2) y = mx + √(a^2m^2 - b^2)
    Normal Equation at (x1, y1) a^2x/x1 - b^2y/y1 = a^2-b^2 a^2x/x1 + b^2y/y1 = a^2-b^2
    Focal Distance Distance from center to vertex > Focal distance Focal distance > Distance from center to vertex
    Asymptotes No Yes
    Eccentricity Always less than 1 Can be greater than 1

    Both ellipses and hyperbolas have vertices, foci, and a center. In the standard equation, both curves have (0, 0) as their center. The main difference between an ellipse and a hyperbola is the shape of the curve: an ellipse is a closed curve, while a hyperbola is an open curve. In addition, ellipses have finite perimeters, whereas hyperbolas have infinite lengths.

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    Guilherme Mazui

    Guilherme Mazui is graduated in journalism from the Federal University of Minas Gerais (UFMG) and a master's degree in Communication from the University of São Paulo (USP). In addition, he has experience in advertising writing and has worked as a content editor in several companies.