An ellipse is orthogonal to the hyperbola x^2-y^2=2 . The...

An ellipse is orthogonal to the hyperbola $x^{2}-y^{2}=2 .$ The eccentricity of the ellipse is reciprocal of that of the hyperbola. Then (a) equation of the ellipse is $x^{2}+2 y^{2}=8$ (b) focus of the ellipse is at $(-4 \sqrt{2}, 0)$ (c) directrix of the ellipse is $x+4 \sqrt{2}=0$ (d) directrix circle of the ellipse $x^{2}+y^{2}=12$

An ellipse is orthogonal to the hyperbola $x^{2}-y^{2}=2 .$ The eccentricity of the ellipse is reciprocal of that of the hyperbola. Then
(a) equation of the ellipse is $x^{2}+2 y^{2}=8$
(b) focus of the ellipse is at $(-4 \sqrt{2}, 0)$
(c) directrix of the ellipse is $x+4 \sqrt{2}=0$
(d) directrix circle of the ellipse $x^{2}+y^{2}=12$

Key Concepts

Orthogonality of Curves

Orthogonality of curves refers to the condition where two curves intersect at right angles, meaning their tangent lines at the point of intersection are perpendicular to each other. In the study of conic sections, this condition provides important constraints on the curves’ parameters. For example, when an ellipse and a hyperbola are orthogonal, relationships between their slopes at the points of intersection can be used to deduce properties such as eccentricity or other parameters.

Directrix

The directrix of a conic section is a fixed line used in one of the definitions of the conic in conjunction with a focus. For any point on the conic, the ratio of its distance from the focus to its distance from the directrix is constant (this constant is the eccentricity). The directrix plays a pivotal role in understanding and defining the geometric properties of conics, including ellipses and hyperbolas.

Reciprocal Relationship of Eccentricities

A reciprocal relationship of eccentricities means that the eccentricity of one conic is the multiplicative inverse of the eccentricity of another. This kind of relationship creates specific constraints on the geometric parameters of the conics involved and is useful in solving problems where the conics' configurations or interactions, such as orthogonality, are considered.

Ellipse

An ellipse is a type of conic section characterized by the set of all points for which the sum of the distances to two fixed points (called foci) is constant. Its standard equation reveals its major and minor axes, and understanding these axes is vital for computing its eccentricity, foci, and directrices. The properties of ellipses are central to many problems in coordinate geometry.

Hyperbola

A hyperbola is a conic section defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. Hyperbolas have characteristic features such as asymptotes, which help describe their behavior at infinity, along with transverse and conjugate axes. Knowledge of these properties is essential for understanding the curve's geometry and its intersections with other conic sections.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a right circular cone. They include ellipses, hyperbolas, parabolas, and circles. Each type has its own standard form and properties, which are crucial for identifying key features such as foci, directrices, axes, and eccentricity. These properties help in understanding the geometric and algebraic nature of these curves.

Eccentricity

Eccentricity is a numerical measure that describes the deviation of a conic section from being a circle. For an ellipse, the eccentricity is between 0 and 1 (with 0 being a circle), while for a hyperbola it is greater than 1. This parameter is crucial in characterizing the shape and nature of the conic, and in many problems, relationships between the eccentricities of different conics are used to derive their equations or other geometric features.

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