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58-562: Marula may refer to: Sclerocarya birrea , a tree native to Africa Marula oil , extracted from the fruits of Sclerocarya birrea Marula, Zimbabwe , a village in Matabeleland South Province Marula mine , an open pit mine in South Africa Marula (poet) (fl. 13th century or earlier), Sanskrit poet from India Topics referred to by

116-454: A a 2 − x 2 = ± ( a 2 − x 2 ) ( 1 − e 2 ) . {\displaystyle y=\pm {\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}=\pm {\sqrt {\left(a^{2}-x^{2}\right)\left(1-e^{2}\right)}}.} The width and height parameters a , b {\displaystyle a,\;b} are called

174-424: 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 e = c a {\displaystyle e={\tfrac {c}{a}}} is the eccentricity . The case F 1 = F 2 {\displaystyle F_{1}=F_{2}} yields

232-528: A 2 x 1 b 2 ) {\displaystyle {\begin{pmatrix}-y_{1}a^{2}&x_{1}b^{2}\end{pmatrix}}} is a tangent vector at point ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} , which proves the vector equation. If ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( u , v ) {\displaystyle (u,v)} are two points of

290-621: A 2 + y 1 v b 2 ) + s 2 ( u 2 a 2 + v 2 b 2 ) = 0   . {\displaystyle {\frac {\left(x_{1}+su\right)^{2}}{a^{2}}}+{\frac {\left(y_{1}+sv\right)^{2}}{b^{2}}}=1\ \quad \Longrightarrow \quad 2s\left({\frac {x_{1}u}{a^{2}}}+{\frac {y_{1}v}{b^{2}}}\right)+s^{2}\left({\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)=0\ .} There are then cases: Using (1) one finds that ( − y 1

348-466: A 2 b 2 . {\displaystyle {\begin{aligned}A&=a^{2}\sin ^{2}\theta +b^{2}\cos ^{2}\theta &B&=2\left(b^{2}-a^{2}\right)\sin \theta \cos \theta \\[1ex]C&=a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta &D&=-2Ax_{\circ }-By_{\circ }\\[1ex]E&=-Bx_{\circ }-2Cy_{\circ }&F&=Ax_{\circ }^{2}+Bx_{\circ }y_{\circ }+Cy_{\circ }^{2}-a^{2}b^{2}.\end{aligned}}} These expressions can be derived from

406-542: A 2 cos 2 ⁡ θ + b 2 sin 2 ⁡ θ D = − 2 A x ∘ − B y ∘ E = − B x ∘ − 2 C y ∘ F = A x ∘ 2 + B x ∘ y ∘ + C y ∘ 2 −

464-462: A 2 − b 2 {\displaystyle c={\sqrt {a^{2}-b^{2}}}} . The eccentricity can be expressed as: e = c a = 1 − ( b a ) 2 , {\displaystyle e={\frac {c}{a}}={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}},} assuming a > b . {\displaystyle a>b.} An ellipse with equal axes (

522-425: A ≥ b > 0   . {\displaystyle a\geq b>0\ .} In principle, the canonical ellipse equation x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} may have a < b {\displaystyle a<b} (and hence

580-458: A + e x {\displaystyle a+ex} and a − e x {\displaystyle a-ex} . It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin. Throughout this article, the semi-major and semi-minor axes are denoted a {\displaystyle a} and b {\displaystyle b} , respectively, i.e.

638-596: A = b {\displaystyle a=b} ) has zero eccentricity, and is a circle. The length of the chord through one focus, perpendicular to the major axis, is called the latus rectum . One half of it is the semi-latus rectum ℓ {\displaystyle \ell } . A calculation shows: ℓ = b 2 a = a ( 1 − e 2 ) . {\displaystyle \ell ={\frac {b^{2}}{a}}=a\left(1-e^{2}\right).} The semi-latus rectum ℓ {\displaystyle \ell }

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696-486: A circle and is included as a special type of ellipse. The equation | P F 2 | + | P F 1 | = 2 a {\displaystyle \left|PF_{2}\right|+\left|PF_{1}\right|=2a} can be viewed in a different way (see figure): c 2 {\displaystyle c_{2}} is called the circular directrix (related to focus F 2 {\displaystyle F_{2}} ) of

754-512: A line outside the ellipse called the directrix : for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity: e = c a = 1 − b 2 a 2 . {\displaystyle e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.} Ellipses are common in physics , astronomy and engineering . For example,

812-522: A point on an ellipse and x → = ( x 1 y 1 ) + s ( u v ) {\textstyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s{\begin{pmatrix}u\\v\end{pmatrix}}} be the equation of any line g {\displaystyle g} containing ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} . Inserting

870-543: A set or locus of points in the Euclidean plane: The midpoint C {\displaystyle C} of the line segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis , and the line perpendicular to it through the center is the minor axis . The major axis intersects the ellipse at two vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} , which have distance

928-440: A strong and distinctive flavour. Inside is a walnut-sized, thick-walled stone ( endocarp ). These stones, when dry, expose the seeds by shedding 2 (sometimes 3) small circular plugs at one end. The fruits are drupes with a single seed encased within their endocarp, although up to four seeds can be present. The seeds have a delicate nutty flavour and are much sought-after, especially by small rodents who know to gnaw exactly where

986-476: Is delivered to processing plants where fruit pulp, pips, kernels and kernel oil are extracted and stored for processing throughout the year. The marula fruit is eaten by various animals in Southern Africa. Giraffes , rhinoceroses and elephants all browse on the marula tree, with elephants in particular being a major consumer. Elephants eat the bark, branches and fruits of the marula, which may limit

1044-422: Is different from Wikidata All article disambiguation pages All disambiguation pages Sclerocarya birrea Sclerocarya birrea ( Ancient Greek : σκληρός ⟨sklērós⟩ , meaning "hard", and κάρυον ⟨káryon⟩ , "nut", in reference to the stone inside the fleshy fruit), commonly known as the marula , is a medium-sized deciduous fruit-bearing tree , indigenous to

1102-569: Is divided into three subspecies: subsp. birrea , subsp. afra and subsp. multifoliolata. These subspecies are differentiated by changes in leaf shape and size. Subsp. birrea is found in northern Africa, subsp. afra is found in southern Africa, and subsp. multifoliolata is only found in Tanzania. The generic name Sclerocarya is derived from the Ancient Greek words 'skleros' meaning 'hard' and 'karyon' meaning 'nut'. This refers to

1160-447: Is equal to the radius of curvature at the vertices (see section curvature ). An arbitrary line g {\displaystyle g} intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line , tangent and secant . Through any point of an ellipse there is a unique tangent. The tangent at a point ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} of

1218-625: Is the 2-argument arctangent function. Using trigonometric functions , a parametric representation of the standard ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} is: ( x , y ) = ( a cos ⁡ t , b sin ⁡ t ) ,   0 ≤ t < 2 π . {\displaystyle (x,\,y)=(a\cos t,\,b\sin t),\ 0\leq t<2\pi \,.} The parameter t (called

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1276-524: Is traditionally used for food in Africa, and has considerable socioeconomic importance. The fruit juice and pulp are mixed with water and stored in a container over 1–3 days of fermentation to make marula beer, a traditional alcoholic beverage . The alcoholic distilled beverage (morula) made from the fruit is referenced in the stories of the South African writer Herman Charles Bosman . Marula oil

1334-521: Is used topically to moisturise the skin, and as an edible oil in the diet of San people in Southern Africa. The marula tree is protected in South Africa. On an industrial level the fruit of the marula tree is collected from the wild by members of rural communities on whose land the trees grow. This harvest and sale of fruit only occur over two to three months, but is an important source of income to poor rural people, especially women. The fruit

1392-805: The eccentric anomaly in astronomy) is not the angle of ( x ( t ) , y ( t ) ) {\displaystyle (x(t),y(t))} with the x -axis, but has a geometric meaning due to Philippe de La Hire (see § Drawing ellipses below). With the substitution u = tan ⁡ ( t 2 ) {\textstyle u=\tan \left({\frac {t}{2}}\right)} and trigonometric formulae one obtains cos ⁡ t = 1 − u 2 1 + u 2   , sin ⁡ t = 2 u 1 + u 2 {\displaystyle \cos t={\frac {1-u^{2}}{1+u^{2}}}\ ,\quad \sin t={\frac {2u}{1+u^{2}}}} and

1450-472: The Bantu in their migrations. There is some evidence of human domestication of marula trees, as trees found on farm lands tend to have larger fruit size. The fruits, which ripen between December and March, have a light yellow skin ( exocarp ), with white flesh ( mesocarp ). They fall to the ground when unripe and green in colour, and then ripen to a yellow colour on the ground. They are succulent and tart with

1508-405: The closed type of conic section : a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas , both of which are open and unbounded . An angled cross section of a right circular cylinder is also an ellipse. An ellipse may also be defined in terms of one focal point and

1566-786: The degenerate cases from the non-degenerate case, let ∆ be the determinant Δ = | A 1 2 B 1 2 D 1 2 B C 1 2 E 1 2 D 1 2 E F | = A C F + 1 4 B D E − 1 4 ( A E 2 + C D 2 + F B 2 ) . {\displaystyle \Delta ={\begin{vmatrix}A&{\frac {1}{2}}B&{\frac {1}{2}}D\\{\frac {1}{2}}B&C&{\frac {1}{2}}E\\{\frac {1}{2}}D&{\frac {1}{2}}E&F\end{vmatrix}}=ACF+{\tfrac {1}{4}}BDE-{\tfrac {1}{4}}(AE^{2}+CD^{2}+FB^{2}).} Then

1624-488: The miombo woodlands of Southern Africa , the Sudano-Sahelian range of West Africa , the savanna woodlands of East Africa and Madagascar . The tree is a single-stemmed species with a broad, spreading crown. It is distinguished by its grey mottled bark and can grow up to 18 meters tall, primarily in low altitudes and open woodlands. The distribution of this species throughout Africa and Madagascar has followed

1682-576: The orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids . A circle viewed from a side angle looks like an ellipse: that is,

1740-491: The radicals by suitable squarings and using b 2 = a 2 − c 2 {\displaystyle b^{2}=a^{2}-c^{2}} (see diagram) produces the standard equation of the ellipse: x 2 a 2 + y 2 b 2 = 1 , {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,} or, solved for y : y = ± b

1798-557: The rational parametric equation of an ellipse { x ( u ) = a 1 − u 2 1 + u 2 y ( u ) = b 2 u 1 + u 2 − ∞ < u < ∞ {\displaystyle {\begin{cases}x(u)=a\,{\dfrac {1-u^{2}}{1+u^{2}}}\\[10mu]y(u)=b\,{\dfrac {2u}{1+u^{2}}}\\[10mu]-\infty <u<\infty \end{cases}}} which covers any point of

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1856-423: The semi-major and semi-minor axes . The top and bottom points V 3 = ( 0 , b ) , V 4 = ( 0 , − b ) {\displaystyle V_{3}=(0,\,b),\;V_{4}=(0,\,-b)} are the co-vertices . The distances from a point ( x , y ) {\displaystyle (x,\,y)} on the ellipse to the left and right foci are

1914-653: The x - and y -axes. In analytic geometry , the ellipse is defined as a quadric : the set of points ( x , y ) {\displaystyle (x,\,y)} of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation A x 2 + B x y + C y 2 + D x + E y + F = 0 {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0} provided B 2 − 4 A C < 0. {\displaystyle B^{2}-4AC<0.} To distinguish

1972-1060: The canonical equation X 2 a 2 + Y 2 b 2 = 1 {\displaystyle {\frac {X^{2}}{a^{2}}}+{\frac {Y^{2}}{b^{2}}}=1} by a Euclidean transformation of the coordinates ( X , Y ) {\displaystyle (X,\,Y)} : X = ( x − x ∘ ) cos ⁡ θ + ( y − y ∘ ) sin ⁡ θ , Y = − ( x − x ∘ ) sin ⁡ θ + ( y − y ∘ ) cos ⁡ θ . {\displaystyle {\begin{aligned}X&=\left(x-x_{\circ }\right)\cos \theta +\left(y-y_{\circ }\right)\sin \theta ,\\Y&=-\left(x-x_{\circ }\right)\sin \theta +\left(y-y_{\circ }\right)\cos \theta .\end{aligned}}} Conversely,

2030-1385: The canonical form parameters can be obtained from the general-form coefficients by the equations: a , b = − 2 ( A E 2 + C D 2 − B D E + ( B 2 − 4 A C ) F ) ( ( A + C ) ± ( A − C ) 2 + B 2 ) B 2 − 4 A C , x ∘ = 2 C D − B E B 2 − 4 A C , y ∘ = 2 A E − B D B 2 − 4 A C , θ = 1 2 atan2 ⁡ ( − B , C − A ) , {\displaystyle {\begin{aligned}a,b&={\frac {-{\sqrt {2{\big (}AE^{2}+CD^{2}-BDE+(B^{2}-4AC)F{\big )}{\big (}(A+C)\pm {\sqrt {(A-C)^{2}+B^{2}}}{\big )}}}}{B^{2}-4AC}},\\x_{\circ }&={\frac {2CD-BE}{B^{2}-4AC}},\\[5mu]y_{\circ }&={\frac {2AE-BD}{B^{2}-4AC}},\\[5mu]\theta &={\tfrac {1}{2}}\operatorname {atan2} (-B,\,C-A),\end{aligned}}} where atan2

2088-456: The ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} except the left vertex ( − a , 0 ) {\displaystyle (-a,\,0)} . For u ∈ [ 0 , 1 ] , {\displaystyle u\in [0,\,1],} this formula represents

2146-479: The ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} has the coordinate equation: x 1 a 2 x + y 1 b 2 y = 1. {\displaystyle {\frac {x_{1}}{a^{2}}}x+{\frac {y_{1}}{b^{2}}}y=1.} A vector parametric equation of

2204-620: The ellipse is a non-degenerate real ellipse if and only if C∆ < 0. If C∆ > 0, we have an imaginary ellipse, and if ∆ = 0, we have a point ellipse. The general equation's coefficients can be obtained from known semi-major axis a {\displaystyle a} , semi-minor axis b {\displaystyle b} , center coordinates ( x ∘ , y ∘ ) {\displaystyle \left(x_{\circ },\,y_{\circ }\right)} , and rotation angle θ {\displaystyle \theta } (the angle from

2262-448: The ellipse is the image of a circle under parallel or perspective projection . The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics . The name, ἔλλειψις ( élleipsis , "omission"), was given by Apollonius of Perga in his Conics . An ellipse can be defined geometrically as

2320-427: The ellipse such that x 1 u a 2 + y 1 v b 2 = 0 {\textstyle {\frac {x_{1}u}{a^{2}}}+{\tfrac {y_{1}v}{b^{2}}}=0} , then the points lie on two conjugate diameters (see below ). (If a = b {\displaystyle a=b} , the ellipse is a circle and "conjugate" means "orthogonal".) If

2378-418: The ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names x {\displaystyle x} and y {\displaystyle y} and the parameter names a {\displaystyle a} and b . {\displaystyle b.} This is the distance from the center to a focus: c =

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2436-543: The ellipse, the x -axis is the major axis, and: For an arbitrary point ( x , y ) {\displaystyle (x,y)} the distance to the focus ( c , 0 ) {\displaystyle (c,0)} is ( x − c ) 2 + y 2 {\textstyle {\sqrt {(x-c)^{2}+y^{2}}}} and to the other focus ( x + c ) 2 + y 2 {\textstyle {\sqrt {(x+c)^{2}+y^{2}}}} . Hence

2494-469: The ellipse. This property should not be confused with the definition of an ellipse using a directrix line below. Using Dandelin spheres , one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone. The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of

2552-415: The flies that have self-domesticated to live near to humans. The ancestral fruit flies are triggered by the ester ethyl isovalerate in the marula fruit. Ellipse An ellipse has a simple algebraic solution for its area, but for its perimeter (also known as circumference ), integration is required to obtain an exact solution. Analytically , the equation of a standard ellipse centered at

2610-588: The foci are ( ± c , 0 ) {\displaystyle (\pm c,0)} for c = a 2 − b 2 {\textstyle c={\sqrt {a^{2}-b^{2}}}} . The standard parametric equation is: ( x , y ) = ( a cos ⁡ ( t ) , b sin ⁡ ( t ) ) for 0 ≤ t ≤ 2 π . {\displaystyle (x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .} Ellipses are

2668-548: The hard pit of the fruit. The specific epithet 'birrea' comes from the common name 'birr', for this type of tree in Senegal . The marula belongs to the same family, Anacardiaceae , as the mango , cashew , pistachio and sumac , and is closely related to the genus Poupartia from Madagascar. Common names include jelly plum, cat thorn, morula, cider tree, marula, maroola nut/plum, and in Afrikaans , maroela. The fruit

2726-558: The line's equation into the ellipse equation and respecting x 1 2 a 2 + y 1 2 b 2 = 1 {\textstyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {y_{1}^{2}}{b^{2}}}=1} yields: ( x 1 + s u ) 2 a 2 + ( y 1 + s v ) 2 b 2 = 1   ⟹ 2 s ( x 1 u

2784-413: The origin with width 2 a {\displaystyle 2a} and height 2 b {\displaystyle 2b} is: x 2 a 2 + y 2 b 2 = 1. {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.} Assuming a ≥ b {\displaystyle a\geq b} ,

2842-737: The parameter [ u : v ] {\displaystyle [u:v]} is considered to be a point on the real projective line P ( R ) {\textstyle \mathbf {P} (\mathbf {R} )} , then the corresponding rational parametrization is [ u : v ] ↦ ( a v 2 − u 2 v 2 + u 2 , b 2 u v v 2 + u 2 ) . {\displaystyle [u:v]\mapsto \left(a{\frac {v^{2}-u^{2}}{v^{2}+u^{2}}},b{\frac {2uv}{v^{2}+u^{2}}}\right).} Then [ 1 : 0 ] ↦ ( −

2900-485: The plugs are located. The trees are dioecious . Male trees produce multiple male flowers on a terminal raceme . These have red sepals and petals, and about 20 stamens per flower. On rare occasion a male flower can produce a gynoecium , turning it bisexual. Female flowers grow individually on their own pedicel and have staminodes . The leaves are alternate, compound, and imparipinnately divided. The leaflet shapes range from round to elliptical . Sclerocarya birrea

2958-399: The point ( x , y ) {\displaystyle (x,\,y)} is on the ellipse whenever: ( x − c ) 2 + y 2 + ( x + c ) 2 + y 2 = 2 a   . {\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}+{\sqrt {(x+c)^{2}+y^{2}}}=2a\ .} Removing

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3016-411: The popular press. While the fruit is commonly eaten by elephants, the animals would need a huge amount of fermented marulas to have any effect on them, and other animals prefer the ripe fruit. The marula fruit has been suggested to be the food of choice for the ancestral forest-dwelling form of the fruit fly Drosophila melanogaster , which was much more selective about which fruit they preferred than

3074-429: The positive horizontal axis to the ellipse's major axis) using the formulae: A = a 2 sin 2 ⁡ θ + b 2 cos 2 ⁡ θ B = 2 ( b 2 − a 2 ) sin ⁡ θ cos ⁡ θ C =

3132-432: The right upper quarter of the ellipse moving counter-clockwise with increasing u . {\displaystyle u.} The left vertex is the limit lim u → ± ∞ ( x ( u ) , y ( u ) ) = ( − a , 0 ) . {\textstyle \lim _{u\to \pm \infty }(x(u),\,y(u))=(-a,\,0)\;.} Alternately, if

3190-408: The same term [REDACTED] This disambiguation page lists articles associated with the title Marula . 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=Marula&oldid=972073037 " Category : Disambiguation pages Hidden categories: Short description

3248-432: The spread of the trees. The damaged bark, due to browsing, can be used to identify marula trees as elephants preferentially target them. Elephants distribute marula seeds in their dung. In the documentary Animals Are Beautiful People by Jamie Uys , released in 1974, some scenes portray elephants , ostriches , warthogs and baboons allegedly becoming intoxicated from eating fermented marula fruit, as do reports in

3306-618: The standard ellipse is shifted to have center ( x ∘ , y ∘ ) {\displaystyle \left(x_{\circ },\,y_{\circ }\right)} , its equation is ( x − x ∘ ) 2 a 2 + ( y − y ∘ ) 2 b 2 = 1   . {\displaystyle {\frac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\frac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1\ .} The axes are still parallel to

3364-606: The tangent is: x → = ( x 1 y 1 ) + s ( − y 1 a 2 x 1 b 2 ) , s ∈ R . {\displaystyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s\left({\begin{array}{r}-y_{1}a^{2}\\x_{1}b^{2}\end{array}}\right),\quad s\in \mathbb {R} .} Proof: Let ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} be

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