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Conjugate axis of a hyperbola

Conjugate Con"ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con- + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See Join.] 1. United in pairs; yoked together; coupled. 2. (Bot.) In single pairs; coupled. 3. (Chem.) Containing two or more radicals supposed to act the part of a single one. [R.] 4. (Gram.) Agreeing in derivation and radical signification; -- said of words. 5. (Math.) Presenting themselves simultaneously and having reciprocal properties; -- frequently used in pure and applied mathematics with reference to two quantities, points, lines, axes, curves, etc. Conjugate axis of a hyperbola (Math.), the line through the center of the curve, perpendicular to the line through the two foci. Conjugate diameters (Conic Sections), two diameters of an ellipse or hyperbola such that each bisects all chords drawn parallel to the other. Conjugate focus (Opt.) See under Focus. Conjugate mirrors (Optics), two mirrors so placed that rays from the focus of one are received at the focus of the other, especially two concave mirrors so placed that rays proceeding from the principal focus of one and reflected in a parallel beam are received upon the other and brought to the principal focus. Conjugate point (Geom.), an acnode. See Acnode, and Double point. Self-conjugate triangle (Conic Sections), a triangle each of whose vertices is the pole of the opposite side with reference to a conic.

Conjugate Con"ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con- + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See Join.] 1. United in pairs; yoked together; coupled. 2. (Bot.) In single pairs; coupled. 3. (Chem.) Containing two or more radicals supposed to act the part of a single one. [R.] 4. (Gram.) Agreeing in derivation and radical signification; -- said of words. 5. (Math.) Presenting themselves simultaneously and having reciprocal properties; -- frequently used in pure and applied mathematics with reference to two quantities, points, lines, axes, curves, etc. Conjugate axis of a hyperbola (Math.), the line through the center of the curve, perpendicular to the line through the two foci. Conjugate diameters (Conic Sections), two diameters of an ellipse or hyperbola such that each bisects all chords drawn parallel to the other. Conjugate focus (Opt.) See under Focus. Conjugate mirrors (Optics), two mirrors so placed that rays from the focus of one are received at the focus of the other, especially two concave mirrors so placed that rays proceeding from the principal focus of one and reflected in a parallel beam are received upon the other and brought to the principal focus. Conjugate point (Geom.), an acnode. See Acnode, and Double point. Self-conjugate triangle (Conic Sections), a triangle each of whose vertices is the pole of the opposite side with reference to a conic.

- the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes. Since both the transverse axis and the conjugate axis are axes of symmetry...

- unit hyperbola requires the conjugate hyperbola y 2 − x 2 = 1 {\displaystyle y^{2}-x^{2}=1} to complement it in the plane. This pair of hyperbolas share...

- a = ℓ e 2 − 1 . {\displaystyle a={\ell \over e^{2}-1}.} The transverse axis of a hyperbola coincides with the major axis. In a hyperbola, a conjugate...

- ellipsoid containing point V are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at V. If one allows the center V to...

- parallel, a single line (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines)...

- section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The...

- ellipse at the endpoint of one diameter is parallel to the conjugate diameter. The longest diameter is called the major axis. The word "diameter" is derived...

- Given a hyperbola with asymptote A, its reflection in A produces the conjugate hyperbola. Any diameter of the original hyperbola is reflected to a conjugate...

- parabola subtends a right angle. The transverse axis of a hyperbola is perpendicular to the conjugate axis and to each directrix. The product of the perpendicular...

- are complex conjugate. An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate...

- unit hyperbola requires the conjugate hyperbola y 2 − x 2 = 1 {\displaystyle y^{2}-x^{2}=1} to complement it in the plane. This pair of hyperbolas share...

- a = ℓ e 2 − 1 . {\displaystyle a={\ell \over e^{2}-1}.} The transverse axis of a hyperbola coincides with the major axis. In a hyperbola, a conjugate...

- ellipsoid containing point V are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at V. If one allows the center V to...

- parallel, a single line (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines)...

- section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The...

- ellipse at the endpoint of one diameter is parallel to the conjugate diameter. The longest diameter is called the major axis. The word "diameter" is derived...

- Given a hyperbola with asymptote A, its reflection in A produces the conjugate hyperbola. Any diameter of the original hyperbola is reflected to a conjugate...

- parabola subtends a right angle. The transverse axis of a hyperbola is perpendicular to the conjugate axis and to each directrix. The product of the perpendicular...

- are complex conjugate. An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate...

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