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73-437: QMS may refer to: Quadrupole mass spectrometer , a scientific instrument Quality management system Quality Meat Scotland Quality Micro Systems , a company founded in 1979 and merged with Minolta's printer division in 2000 Quantum Markov semigroup  – A kind of mathematical structure which describes the dynamics in a Markovian open quantum system. Minolta-QMS ,

146-437: A {\displaystyle \left|\left|PF_{2}\right|-\left|PF_{1}\right|\right|=2a} can be viewed in a different way (see diagram): If c 2 {\displaystyle c_{2}} is the circle with midpoint F 2 {\displaystyle F_{2}} and radius 2 a {\displaystyle 2a} , then the distance of a point P {\displaystyle P} of

219-496: A {\displaystyle a} to the center. The distance c {\displaystyle c} of the foci to the center is called the focal distance or linear eccentricity . The quotient c a {\displaystyle {\tfrac {c}{a}}} is the eccentricity e {\displaystyle e} . The equation | | P F 2 | − | P F 1 | | = 2

292-394: A ) 2 a 2 − y 2 b 2 = 1 , {\displaystyle {\frac {(x+a)^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1\,,} which is the equation of a hyperbola with center ( − a , 0 ) {\displaystyle (-a,0)} , the x -axis as major axis and the major/minor semi axis

365-457: A 2 x 2 − b 2 {\displaystyle y^{2}={\tfrac {b^{2}}{a^{2}}}x^{2}-b^{2}} satisfy the equation | P F 1 | 2 − c 2 a 2 | P l 1 | 2 = 0   . {\displaystyle |PF_{1}|^{2}-{\frac {c^{2}}{a^{2}}}|Pl_{1}|^{2}=0\ .} The second case

438-423: A 2 − y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1} one uses the pencils at the vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} . Let P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})} be

511-391: A 2 = 1 {\displaystyle {\tfrac {x^{2}-y^{2}}{a^{2}}}=1} (whose semi-axes are equal) has the new equation 2 ξ η a 2 = 1 {\displaystyle {\tfrac {2\xi \eta }{a^{2}}}=1} . Solving for η {\displaystyle \eta } yields η =

584-481: A 2 / 2 ξ   . {\displaystyle \eta ={\tfrac {a^{2}/2}{\xi }}\ .} Thus, in an xy -coordinate system the graph of a function f : x ↦ A x , A > 0 , {\displaystyle f:x\mapsto {\tfrac {A}{x}},\;A>0\;,} with equation y = A x , A > 0 , {\displaystyle y={\frac {A}{x}}\;,A>0\;,}

657-400: A , b {\displaystyle a,b} so that e 2 − 1 = b 2 a 2 ,  and    p = b 2 a {\displaystyle e^{2}-1={\tfrac {b^{2}}{a^{2}}},{\text{ and }}\ p={\tfrac {b^{2}}{a}}} , and then the equation above becomes ( x +

730-468: A , b {\displaystyle a,b} . Because of c ⋅ a 2 c = a 2 {\displaystyle c\cdot {\tfrac {a^{2}}{c}}=a^{2}} point L 1 {\displaystyle L_{1}} of directrix l 1 {\displaystyle l_{1}} (see diagram) and focus F 1 {\displaystyle F_{1}} are inverse with respect to

803-404: A , b , p , c , e {\displaystyle a,b,p,c,e} remain unchanged. The two lines at distance d = a 2 c {\textstyle d={\frac {a^{2}}{c}}} from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram). For an arbitrary point P {\displaystyle P} of

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876-599: A parabola and if e < 1 {\displaystyle e<1} an ellipse .) Let F = ( f , 0 ) ,   e > 0 {\displaystyle F=(f,0),\ e>0} and assume ( 0 , 0 ) {\displaystyle (0,0)} is a point on the curve. The directrix l {\displaystyle l} has equation x = − f e {\displaystyle x=-{\tfrac {f}{e}}} . With P = ( x , y ) {\displaystyle P=(x,y)} ,

949-408: A hybrid instrument. A mass-selecting quadrupole and collision quadrupole with time-of-flight device as the second mass selection stage is a hybrid known as a quadrupole time-of-flight mass spectrometer (QTOF MS). Quadrupole-quadrupole-time-of-flight (QqTOF) configurations are also possible and used especially the mass spectrometry of peptides and other large biological polymers. A variant of

1022-402: A hyperbola can arise as the path followed by the shadow of the tip of a sundial 's gnomon , the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle , among others. Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from

1095-509: A hyperbola in this section gives a parametric representation of an arbitrary hyperbola, even in space, if one allows f → 0 , f → 1 , f → 2 {\displaystyle {\vec {f}}\!_{0},{\vec {f}}\!_{1},{\vec {f}}\!_{2}} to be vectors in space. Because the unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1}

1168-461: A hyperbola uses affine transformations : An affine transformation of the Euclidean plane has the form x → → f → 0 + A x → {\displaystyle {\vec {x}}\to {\vec {f}}_{0}+A{\vec {x}}} , where A {\displaystyle A} is a regular matrix (its determinant

1241-753: A method of regulating catch in a fishery Qatar Motor Show , a trade show Schools [ edit ] Qualters Middle School , a grade 6-8 middle school located in Mansfield, Massachusetts Queensborough Middle School , a school in Queensborough , New Westminster, British Columbia, Canada Queen Margaret's School , a private boarding school/Day School, located in Duncan, British Columbia, Canada Quest Middle School, another name for IDEA Quest located in Edinburg, Texas Quinte Mohawk School ,

1314-472: A point of the hyperbola and A = ( a , y 0 ) , B = ( x 0 , 0 ) {\displaystyle A=(a,y_{0}),B=(x_{0},0)} . The line segment B P ¯ {\displaystyle {\overline {BP}}} is divided into n equally-spaced segments and this division is projected parallel with the diagonal A B {\displaystyle AB} as direction onto

1387-526: A primary school in Tyendinaga, Ontario Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title QMS . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=QMS&oldid=1223110772 " Category : Disambiguation pages Hidden categories: Short description

1460-603: A printer company merged into Konica Minolta in 2003 Quantitative Micro Software , a company who developed the EViews software Quartermaster Sergeant , a type of appointment in the British Army and Royal Marines Queue management system Quick Media Switching in HDMI Quicksilver Messenger Service , an American psychedelic rock band of the 1960s Quota Management System ,

1533-2597: A vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter t 0 {\displaystyle t_{0}} of a vertex from the equation p → ′ ( t ) ⋅ ( p → ( t ) − f → 0 ) = ( f → 1 sinh ⁡ t + f → 2 cosh ⁡ t ) ⋅ ( f → 1 cosh ⁡ t + f → 2 sinh ⁡ t ) = 0 {\displaystyle {\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\right)\cdot \left({\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\right)=0} and hence from coth ⁡ ( 2 t 0 ) = − f → 1 2 + f → 2 2 2 f → 1 ⋅ f → 2   , {\displaystyle \coth(2t_{0})=-{\tfrac {{\vec {f}}_{1}^{\,2}+{\vec {f}}_{2}^{\,2}}{2{\vec {f}}_{1}\cdot {\vec {f}}_{2}}}\ ,} which yields t 0 = 1 4 ln ⁡ ( f → 1 − f → 2 ) 2 ( f → 1 + f → 2 ) 2 . {\displaystyle t_{0}={\tfrac {1}{4}}\ln {\tfrac {\left({\vec {f}}_{1}-{\vec {f}}_{2}\right)^{2}}{\left({\vec {f}}_{1}+{\vec {f}}_{2}\right)^{2}}}.} The formulae cosh 2 ⁡ x + sinh 2 ⁡ x = cosh ⁡ 2 x {\displaystyle \cosh ^{2}x+\sinh ^{2}x=\cosh 2x} , 2 sinh ⁡ x cosh ⁡ x = sinh ⁡ 2 x {\displaystyle 2\sinh x\cosh x=\sinh 2x} , and arcoth ⁡ x = 1 2 ln ⁡ x + 1 x − 1 {\displaystyle \operatorname {arcoth} x={\tfrac {1}{2}}\ln {\tfrac {x+1}{x-1}}} were used. The two vertices of

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1606-426: Is a rectangular hyperbola entirely in the first and third quadrants with A rotation of the original hyperbola by − 45 ∘ {\displaystyle -45^{\circ }} results in a rectangular hyperbola entirely in the second and fourth quadrants, with the same asymptotes, center, semi-latus rectum, radius of curvature at the vertices, linear eccentricity, and eccentricity as for

1679-402: Is a point of the hyperbola. The tangent vector is p → ′ ( t ) = f → 1 − f → 2 1 t 2 . {\displaystyle {\vec {p}}'(t)={\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}.} At a vertex the tangent is perpendicular to

1752-830: Is affinely equivalent to the hyperbola y = 1 / x {\displaystyle y=1/x} , an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola y = 1 / x {\displaystyle y=1/x\,} : x → = p → ( t ) = f → 0 + f → 1 t + f → 2 1 t , t ≠ 0 . {\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}+{\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}},\quad t\neq 0\,.} M : f → 0 {\displaystyle M:{\vec {f}}_{0}}

1825-685: Is applied to the instrument. Manifold improvements in resolution and sensitivity have been reported for a magnetic field applied in various orientations to a QMS. These mass spectrometers excel at applications where particular ions of interest are being studied because they can stay tuned on a single ion for extended periods of time. One place where this is useful is in liquid chromatography-mass spectrometry or gas chromatography-mass spectrometry where they serve as exceptionally high specificity detectors. Quadrupole instruments are often reasonably priced and make good multi-purpose instruments. A single quadrupole mass spectrometer with an electron impact ionizer

1898-416: Is called the circular directrix (related to focus F 2 {\displaystyle F_{2}} ) of the hyperbola. In order to get the left branch of the hyperbola, one has to use the circular directrix related to F 1 {\displaystyle F_{1}} . This property should not be confused with the definition of a hyperbola with help of a directrix (line) below. If

1971-405: Is different from Wikidata All article disambiguation pages All disambiguation pages Quadrupole mass spectrometer The quadrupole consists of four parallel metal rods. Each opposing rod pair is connected together electrically, and a radio frequency (RF) voltage with a DC offset voltage is applied between one pair of rods and the other. Ions travel down the quadrupole between

2044-452: Is employed as a collision cell. This collision cell is an RF-only quadrupole (non-mass filtering) using Ar, He, or N 2 gas (~10 Torr, ~30 eV) for collision induced dissociation of selected parent ion(s) from Q 1 . Subsequent fragments are passed through to Q 3 where they may be filtered or fully scanned. This process allows for the study of fragments that are useful in structural elucidation by tandem mass spectrometry . For example,

2117-423: Is equivalent to t 0 = ± 1 {\displaystyle t_{0}=\pm 1} and f → 0 ± ( f → 1 + f → 2 ) {\displaystyle {\vec {f}}_{0}\pm ({\vec {f}}_{1}+{\vec {f}}_{2})} are the vertices of the hyperbola. The following properties of

2190-549: Is mapped onto the hyperbola x → = p → ( t ) = f → 0 ± f → 1 cosh ⁡ t + f → 2 sinh ⁡ t   . {\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}\pm {\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\ .} f → 0 {\displaystyle {\vec {f}}_{0}}

2263-579: Is not Euclidean ). The word "hyperbola" derives from the Greek ὑπερβολή , meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube , but were then called sections of obtuse cones. The term hyperbola is believed to have been coined by Apollonius of Perga ( c.  262  – c.  190 BC ) in his definitive work on

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2336-611: Is not 0) and f → 0 {\displaystyle {\vec {f}}_{0}} is an arbitrary vector. If f → 1 , f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} are the column vectors of the matrix A {\displaystyle A} , the unit hyperbola ( ± cosh ⁡ ( t ) , sinh ⁡ ( t ) ) , t ∈ R , {\displaystyle (\pm \cosh(t),\sinh(t)),t\in \mathbb {R} ,}

2409-409: Is one of the three kinds of conic section , formed by the intersection of a plane and a double cone . (The other conic sections are the parabola and the ellipse . A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Besides being a conic section, a hyperbola can arise as

2482-479: Is proven analogously. The inverse statement is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola): For any point F {\displaystyle F} (focus), any line l {\displaystyle l} (directrix) not through F {\displaystyle F} and any real number e {\displaystyle e} with e > 1 {\displaystyle e>1}

2555-397: Is the center of the hyperbola, the vectors f → 1 , f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} have the directions of the asymptotes and f → 1 + f → 2 {\displaystyle {\vec {f}}_{1}+{\vec {f}}_{2}}

2628-744: Is the center, f → 0 + f → 1 {\displaystyle {\vec {f}}_{0}+{\vec {f}}_{1}} a point of the hyperbola and f → 2 {\displaystyle {\vec {f}}_{2}} a tangent vector at this point. In general the vectors f → 1 , f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} are not perpendicular. That means, in general f → 0 ± f → 1 {\displaystyle {\vec {f}}_{0}\pm {\vec {f}}_{1}} are not

2701-442: Is the equation of an ellipse ( e < 1 {\displaystyle e<1} ) or a parabola ( e = 1 {\displaystyle e=1} ) or a hyperbola ( e > 1 {\displaystyle e>1} ). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram). If e > 1 {\displaystyle e>1} , introduce new parameters

2774-545: Is the intersection of the asymptote with its perpendicular through F 1 {\displaystyle F_{1}} (see diagram). The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two Dandelin spheres d 1 , d 2 {\displaystyle d_{1},d_{2}} , which are spheres that touch

2847-412: Is the perpendicular to line F 1 F 2 ¯ {\displaystyle {\overline {F_{1}F_{2}}}} through point E 1 {\displaystyle E_{1}} . Alternative construction of E 1 {\displaystyle E_{1}} : Calculation shows, that point E 1 {\displaystyle E_{1}}

2920-886: Is the solution of the equation ( y − y 1 ) ( x − x 1 ) ( x − x 2 ) ( y − y 2 ) = ( y 3 − y 1 ) ( x 3 − x 1 ) ( x 3 − x 2 ) ( y 3 − y 2 ) {\displaystyle {\frac {({\color {red}y}-y_{1})}{({\color {green}x}-x_{1})}}{\frac {({\color {green}x}-x_{2})}{({\color {red}y}-y_{2})}}={\frac {(y_{3}-y_{1})}{(x_{3}-x_{1})}}{\frac {(x_{3}-x_{2})}{(y_{3}-y_{2})}}} for y {\displaystyle {\color {red}y}} . Another definition of

2993-492: Is used as a standalone analyzer in residual gas analyzers , real-time gas analyzers, plasma diagnostics and SIMS surface analysis systems. Hyperbola In mathematics , a hyperbola is a type of smooth curve lying in a plane , defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows . The hyperbola

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3066-394: The circle inversion at circle x 2 + y 2 = a 2 {\displaystyle x^{2}+y^{2}=a^{2}} (in diagram green). Hence point E 1 {\displaystyle E_{1}} can be constructed using the theorem of Thales (not shown in the diagram). The directrix l 1 {\displaystyle l_{1}}

3139-507: The conic sections , the Conics . The names of the other two general conic sections, the ellipse and the parabola , derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than

3212-457: The foci of the hyperbola. The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler: The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section : For the generation of points of the hyperbola x 2

3285-399: The locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship x y = 1. {\displaystyle xy=1.} In practical applications,

3358-628: The xy -coordinate system is rotated about the origin by the angle + 45 ∘ {\displaystyle +45^{\circ }} and new coordinates ξ , η {\displaystyle \xi ,\eta } are assigned, then x = ξ + η 2 , y = − ξ + η 2 {\displaystyle x={\tfrac {\xi +\eta }{\sqrt {2}}},\;y={\tfrac {-\xi +\eta }{\sqrt {2}}}} . The rectangular hyperbola x 2 − y 2

3431-472: The 1980s, the MAT company and subsequently Finnigan Instrument Corporation used hyperbolic rods produced with a mechanical tolerance of 0.001 mm, whose exact production process was a well-kept secret within the company. A linear series of three quadrupoles is known as a triple quadrupole mass spectrometer . The first (Q 1 ) and third (Q 3 ) quadrupoles act as mass filters, and the middle (q 2 ) quadrupole

3504-460: The Q 1 may be set to 'filter' for a drug ion of known mass, which is fragmented in q 2 . The third quadrupole (Q 3 ) can then be set to scan the entire m/z range, giving information on the intensities of the fragments. Thus, the structure of the original ion can be deduced. The arrangement of three quadrupoles was first developed by Jim Morrison of La Trobe University in Australia for

3577-639: The asymptotes are the two coordinate axes . Hyperbolas share many of the ellipses' analytical properties such as eccentricity , focus , and directrix . Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry ( Lobachevsky 's celebrated non-Euclidean geometry ), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which

3650-479: The case of + 45 ∘ {\displaystyle +45^{\circ }} rotation, with equation y = − A x ,     A > 0 , {\displaystyle y=-{\frac {A}{x}}\;,~~A>0\;,} Shifting the hyperbola with equation y = A x ,   A ≠ 0   , {\displaystyle y={\frac {A}{x}},\ A\neq 0\ ,} so that

3723-463: The center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve y ( x ) = 1 / x {\displaystyle y(x)=1/x}

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3796-441: The cone along circles c 1 {\displaystyle c_{1}} , c 2 {\displaystyle c_{2}} and the intersecting (hyperbola) plane at points F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} . It turns out: F 1 , F 2 {\displaystyle F_{1},F_{2}} are

3869-1226: The following statement is true: The four points are on a hyperbola with equation y = a x − b + c {\displaystyle y={\tfrac {a}{x-b}}+c} if and only if the angles at P 3 {\displaystyle P_{3}} and P 4 {\displaystyle P_{4}} are equal in the sense of the measurement above. That means if ( y 4 − y 1 ) ( x 4 − x 1 ) ( x 4 − x 2 ) ( y 4 − y 2 ) = ( y 3 − y 1 ) ( x 3 − x 1 ) ( x 3 − x 2 ) ( y 3 − y 2 ) {\displaystyle {\frac {(y_{4}-y_{1})}{(x_{4}-x_{1})}}{\frac {(x_{4}-x_{2})}{(y_{4}-y_{2})}}={\frac {(y_{3}-y_{1})}{(x_{3}-x_{1})}}{\frac {(x_{3}-x_{2})}{(y_{3}-y_{2})}}} The proof can be derived by straightforward calculation. If

3942-399: The hyperbola are f → 0 ± ( f → 1 cosh ⁡ t 0 + f → 2 sinh ⁡ t 0 ) . {\displaystyle {\vec {f}}_{0}\pm \left({\vec {f}}_{1}\cosh t_{0}+{\vec {f}}_{2}\sinh t_{0}\right).} Solving

4015-505: The hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: | P F 1 | | P l 1 | = | P F 2 | | P l 2 | = e = c a . {\displaystyle {\frac {|PF_{1}|}{|Pl_{1}|}}={\frac {|PF_{2}|}{|Pl_{2}|}}=e={\frac {c}{a}}\,.} The proof for

4088-580: The line segment A P ¯ {\displaystyle {\overline {AP}}} (see diagram). The parallel projection is part of the projective mapping between the pencils at V 1 {\displaystyle V_{1}} and V 2 {\displaystyle V_{2}} needed. The intersection points of any two related lines S 1 A i {\displaystyle S_{1}A_{i}} and S 2 B i {\displaystyle S_{2}B_{i}} are points of

4161-982: The major axis. Hence p → ′ ( t ) ⋅ ( p → ( t ) − f → 0 ) = ( f → 1 − f → 2 1 t 2 ) ⋅ ( f → 1 t + f → 2 1 t ) = f → 1 2 t − f → 2 2 1 t 3 = 0 {\displaystyle {\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}\right)\cdot \left({\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}\right)={\vec {f}}_{1}^{2}t-{\vec {f}}_{2}^{2}{\tfrac {1}{t^{3}}}=0} and

4234-513: The new center is ( c 0 , d 0 ) {\displaystyle (c_{0},d_{0})} , yields the new equation y = A x − c 0 + d 0 , {\displaystyle y={\frac {A}{x-c_{0}}}+d_{0}\;,} and the new asymptotes are x = c 0 {\displaystyle x=c_{0}} and y = d 0 {\displaystyle y=d_{0}} . The shape parameters

4307-590: The pair F 1 , l 1 {\displaystyle F_{1},l_{1}} follows from the fact that | P F 1 | 2 = ( x − c ) 2 + y 2 ,   | P l 1 | 2 = ( x − a 2 c ) 2 {\displaystyle |PF_{1}|^{2}=(x-c)^{2}+y^{2},\ |Pl_{1}|^{2}=\left(x-{\tfrac {a^{2}}{c}}\right)^{2}} and y 2 = b 2

4380-531: The parameter of a vertex is t 0 = ± f → 2 2 f → 1 2 4 . {\displaystyle t_{0}=\pm {\sqrt[{4}]{\frac {{\vec {f}}_{2}^{2}}{{\vec {f}}_{1}^{2}}}}.} | f → 1 | = | f → 2 | {\displaystyle \left|{\vec {f}}\!_{1}\right|=\left|{\vec {f}}\!_{2}\right|}

4453-1195: The parametric representation for cosh ⁡ t , sinh ⁡ t {\displaystyle \cosh t,\sinh t} by Cramer's rule and using cosh 2 ⁡ t − sinh 2 ⁡ t − 1 = 0 {\displaystyle \;\cosh ^{2}t-\sinh ^{2}t-1=0\;} , one gets the implicit representation det ( x → − f → 0 , f → 2 ) 2 − det ( f → 1 , x → − f → 0 ) 2 − det ( f → 1 , f → 2 ) 2 = 0. {\displaystyle \det \left({\vec {x}}\!-\!{\vec {f}}\!_{0},{\vec {f}}\!_{2}\right)^{2}-\det \left({\vec {f}}\!_{1},{\vec {x}}\!-\!{\vec {f}}\!_{0}\right)^{2}-\det \left({\vec {f}}\!_{1},{\vec {f}}\!_{2}\right)^{2}=0.} The definition of

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4526-709: The points are on a hyperbola, one can assume the hyperbola's equation is y = a / x {\displaystyle y=a/x} . A consequence of the inscribed angle theorem for hyperbolas is the 3-point-form of a hyperbola's equation  —  The equation of the hyperbola determined by 3 points P i = ( x i , y i ) ,   i = 1 , 2 , 3 ,   x i ≠ x k , y i ≠ y k , i ≠ k {\displaystyle P_{i}=(x_{i},y_{i}),\ i=1,2,3,\ x_{i}\neq x_{k},y_{i}\neq y_{k},i\neq k}

4599-405: The purpose of studying the photodissociation of gas-phase ions. The first triple-quadrupole mass spectrometer was developed at Michigan State University by Christie Enke and graduate student Richard Yost in the late 1970s. Quadrupoles can be used in hybrid mass spectrometers . For example, a sector instrument can be combined with a collision quadrupole and quadrupole mass analyzer to form

4672-415: The quadrupole mass analyzer called the monopole was invented by von Zahn which operates with two electrodes and generates one quarter of the quadrupole field. It has one circular electrode and one V-shaped electrode. The performance is, however, lower than that of a quadrupole mass analyzer. An enhancement to the performance of the quadrupole mass analyzer has been demonstrated to occur when a magnetic field

4745-513: The relation | P F | 2 = e 2 | P l | 2 {\displaystyle |PF|^{2}=e^{2}|Pl|^{2}} produces the equations The substitution p = f ( 1 + e ) {\displaystyle p=f(1+e)} yields x 2 ( e 2 − 1 ) + 2 p x − y 2 = 0. {\displaystyle x^{2}(e^{2}-1)+2px-y^{2}=0.} This

4818-393: The right branch to the circle c 2 {\displaystyle c_{2}} equals the distance to the focus F 1 {\displaystyle F_{1}} : | P F 1 | = | P c 2 | . {\displaystyle |PF_{1}|=|Pc_{2}|.} c 2 {\displaystyle c_{2}}

4891-404: The rods are hyperbolic , however cylindrical rods with a specific ratio of rod diameter-to-spacing provide an easier-to-manufacture adequate approximation to hyperbolas. Small variations in the ratio have large effects on resolution and peak shape. Different manufacturers choose slightly different ratios to fine-tune operating characteristics in context of anticipated application requirements. Since

4964-495: The rods. Only ions of a certain mass-to-charge ratio will reach the detector for a given ratio of voltages: other ions have unstable trajectories and will collide with the rods. This permits selection of an ion with a particular m/z or allows the operator to scan for a range of m/z -values by continuously varying the applied voltage. Mathematically this can be modeled with the help of the Mathieu differential equation . Ideally,

5037-477: The segment or exceed the segment. A hyperbola can be defined geometrically as a set of points ( locus of points ) in the Euclidean plane: The midpoint M {\displaystyle M} of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis . It contains the vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} , which have distance

5110-474: The set of points (locus of points), for which the quotient of the distances to the point and to the line is e {\displaystyle e} H = { P | | P F | | P l | = e } {\displaystyle H=\left\{P\,{\Biggr |}\,{\frac {|PF|}{|Pl|}}=e\right\}} is a hyperbola. (The choice e = 1 {\displaystyle e=1} yields

5183-754: The shape parameters a , b , c {\displaystyle a,b,c} uses the inscribed angle theorem for hyperbolas: Analogous to the inscribed angle theorem for circles one gets the Inscribed angle theorem for hyperbolas  —  For four points P i = ( x i , y i ) ,   i = 1 , 2 , 3 , 4 ,   x i ≠ x k , y i ≠ y k , i ≠ k {\displaystyle P_{i}=(x_{i},y_{i}),\ i=1,2,3,4,\ x_{i}\neq x_{k},y_{i}\neq y_{k},i\neq k} (see diagram)

5256-603: The uniquely defined hyperbola. Remarks: A hyperbola with equation y = a x − b + c ,   a ≠ 0 {\displaystyle y={\tfrac {a}{x-b}}+c,\ a\neq 0} is uniquely determined by three points ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) {\displaystyle (x_{1},y_{1}),\;(x_{2},y_{2}),\;(x_{3},y_{3})} with different x - and y -coordinates. A simple way to determine

5329-714: The vertices of the hyperbola. But f → 1 ± f → 2 {\displaystyle {\vec {f}}_{1}\pm {\vec {f}}_{2}} point into the directions of the asymptotes. The tangent vector at point p → ( t ) {\displaystyle {\vec {p}}(t)} is p → ′ ( t ) = f → 1 sinh ⁡ t + f → 2 cosh ⁡ t   . {\displaystyle {\vec {p}}'(t)={\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\ .} Because at

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